Faculté des Sciences
Département de Mathématique
Connecting hitting sets and hitting paths
in graphs
Eglantine CAMBY
Thèse présentée en vue de l’obtention du grade de Docteur en Sciences
UNIVERSITÉ LIBRE DE BRUXELLES
Faculté des Sciences
Département de Mathématique
Connecting hitting sets and hitting paths
in graphs
Eglantine CAMBY
Thèse présentée en vue de l’obtention du grade de Docteur en Sciences
Juin 2015
Jury de Thèse
G. Joret (président), Université libre de Bruxelles, Belgium
S. Fiorini (promoteur), Université libre de Bruxelles, Belgium
J. Cardinal (co-promoteur & secrétaire), Univ. libre de Bruxelles, Belgium
A. Hertz, Polytechnique Montréal, Canada
O. Schaudt, Universität zu Köln, Germany
I. Todinca, Université d’Orléans, France
“Hard try and never give up”
Author Unknown.
“All things are difficult before they are easy”
Thomas Fuller.
Remerciements
Après cinq années de thèse, je tiens à remercier les personnes qui m’ont
accompagnée et soutenue durant cette longue période.
Cette thèse a vu le jour sous la direction de Samuel Fiorini, mon promoteur. Je tiens à le remercier de m’avoir suivi dans ce projet, d’avoir cru en
mes idées et de m’avoir laissé suffisamment de liberté pour découvrir l’univers
scientifique à travers le monde. Ce premier a essentiellement mis l’accent sur
le côté structurel de la recherche.
Par la suite, j’ai eu l’honneur d’avoir Jean Cardinal comme co-promoteur.
Étant informaticien, Jean Cardinal a attiré mon attention sur le côté algorithmique de la recherche. Je tiens à le remercier de m’avoir initié, avec
Hadrien Mélot, à la recherche via le mémoire de Master, et aussi de m’avoir
guidé à travers ces années.
Ce tandem est d’abord une richesse, ensuite une occasion d’apprendre
deux fois plus. Il m’a donné le moyen de me forger ma propre opinion sur
base de leurs opinions, souvent complémentaires. Ce duo m’a enseigné une
précision d’écriture tant au point de vue du contenu que de la forme.
Je tiens à remercier Gwenaël Joret, Alain Hertz, Oliver Schaudt et Ioan
Todinca pour avoir accepté de faire partie de mon jury de thèse et pour
leur lecture attentive de ma thèse ainsi que leurs commentaires. I would
like to address special thanks to Oliver Schaudt for his kind and rewarding
collaboration. De même, j’aimerais remercier tout particulièrement Alain
Hertz et Hadrien Mélot pour m’avoir appris à accorder du crédit à mes
idées.
Je remercie l’ensemble des professeurs de l’UMons pour m’avoir initié aux
mathématiques et pour m’avoir donné le goût de la recherche. J’adresse mes
remerciements à Olivier et Stéphanie pour m’avoir épaulé dans des moments
de tristesse.
J’aimerais aussi remercier les personnes qui m’ont aidée avec des relectures tant au niveau scientifique qu’au niveau linguistique.
Je remercie également le Fonds de la Recherche Scientifique, l’Université
Libre de Bruxelles, la Fédération Wallonie-Bruxelles ainsi que mes deux promoteurs pour le soutien financier apporté durant mes nombreuses missions
scientifiques à travers le monde.
Partager le Savoir est primordial. Avec ce but, j’ai pu m’épanouir dans
i
ii
le métier d’assistante. Enseigner est une expérience enrichissante sur le plan
intellectuel mais aussi humain. Je remercie tous mes étudiants, en particulier, Bob, Fränk, Mélanie, Alex, Cédric, Blaise, Vincent, Sofiane, Lancelot,
Matvei, Philippe, Corentin, Bálint, . . . Je tiens spécialement à remercier Yan
et Georges pour leur soutien dans les moments difficiles mais aussi Eileen
dont le chemin est intimement lié au mien.
Ces cinq dernières années n’ont pas toujours été dans la joie. Cependant,
j’ai rencontré une personne formidable durant mes premiers cours du soir
d’anglais : Seher, ma meilleure amie. Même si les maths sont pour elle
comme une langue étrangère, Seher m’a toujours soutenue dans mes décisions
et m’a toujours entourée de son amour inconditionnel. Peu importe l’endroit,
peu importe le moment, je peux compter sur elle. Merci with love, Seher, de
m’avoir accompagné dans ces moments agréables et/ou difficiles.
Parmi mes collègues, je remercie Thomas, Selim, Rémi, Robson, Nathann
et Audrey. J’ai pu particulièrement compter sur Sarah et Carine, alias my
Cherry et Mademoiselle, avec qui j’ai grandi, rigolé mais aussi pleuré. Je les
remercie infininement pour leur gentillesse et leur amour.
De plus, je tiens spécialement à remercier Jean-Paul. Je n’aurais pas pu
espérer un meilleur supérieur que Jean-Paul. Merci pour ta bienveillance et
ton amabilité.
Parmi mes artistes favoris, je cite Nadège. Merci Nadj de m’avoir ouvert
ta porte lorsque j’en avais le plus besoin. Merci pour ta bonne humeur et
tes délires. Merci de m’avoir parlé au bord de la piscine de Soignies il y a
maintenant huit ans.
Je remercie également Alain, un autre de mes artistes favoris. Il m’a
ouvert au monde du spectacle et restera probablement un de mes meilleurs
fans. Merci de garder un telle admiration à mon égard.
Merci à Marie-Eve, Italia et Nadir pour tous les rayons de soleil envoyés
depuis tant d’années.
Depuis l’autre bout du monde, je remercie Corine de m’avoir accueilli
si chaleureusement à Montréal et de m’avoir encouragé lorsque le doute
m’envahissait.
J’adresse un tout grand merci à chacun des membres de ma famille :
Papa, Maman, Gabriel, Anémone, Amélie, Pierre. Chacun de vous m’a
aidée à sa façon à garder le cap dans ma vie, et en particulier dans ce projet
qu’est la thèse. Merci à Parrain, Charlotte et leurs enfants pour m’avoir
accueilli dans leur maison comme si j’étais leur propre enfant.
Je tiens aussi à remercier Dora. Merci pour ces sept années de vie commune, merci de me câliner lorsque mon moral est au plus bas, merci de veiller
avec moi durant ces longues nuits de travail, merci d’avoir partagé ce stress
qui m’habitait.
Finalement, je remercie Gilles pour son amour inconditionnel, son soutien
constant et sa connexion exceptionnelle, grâce auxquels j’ai pu aller au bout
d’un de mes rêves.
Contents
1 Introduction
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Background
2.1 Computational complexity . . . . . . . .
2.1.1 Classes P and N P . . . . . . . .
2.1.2 Class Θp2 . . . . . . . . . . . . . .
2.2 Approximation algorithms . . . . . . . .
2.3 Graphs and hypergraphs . . . . . . . . .
2.4 H -hitting set problems . . . . . . . . .
2.4.1 The vertex cover problem . . . .
2.4.2 The dominating set problem . . .
2.4.3 The feedback vertex set problem
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3 The price of connectivity and other prices
3.1 Famous prices . . . . . . . . . . . . . . . . . . . . . . .
3.2 Price of connectivity . . . . . . . . . . . . . . . . . . .
3.2.1 Vertex cover problem . . . . . . . . . . . . . . .
3.2.2 Dominating set problem . . . . . . . . . . . . .
3.2.3 Feedback vertex set problem . . . . . . . . . . .
3.3 Other prices . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Paired-domination versus total domination . .
3.3.2 Connected domination versus total domination
4 The price of connectivity for vertex cover
4.1 Computational Complexity . . . . . . . . . . .
4.2 Structural results . . . . . . . . . . . . . . . . .
4.2.1 PoC-Perfect Graphs . . . . . . . . . . .
4.2.2 PoC-Near-Perfect Graphs . . . . . . . .
4.2.3 PoC-Critical Graphs . . . . . . . . . . .
4.2.4 Aside: values of the price of connectivity
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5 The price of connectivity for domination
5.1 Computational Complexity . . . . . . . .
5.2 Structural results . . . . . . . . . . . . . .
5.2.1 PoC-Near-Perfect Graphs . . . . .
5.2.2 PoC-Critical Graphs . . . . . . . .
5.2.3 Aside . . . . . . . . . . . . . . . .
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6 A characterization of Pk -free graphs
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6.1 A characterization of Pk -free graphs . . . . . . . . . . . . . . 75
6.2 A polynomial-time algorithm to find a special connected dominating set in Pk -free graphs . . . . . . . . . . . . . . . . . . . 77
7 2-colorability of hypergraphs
7.1 Classical results on hypergraph colorability
7.1.1 Sufficient conditions . . . . . . . . .
7.1.2 Computational complexity results . .
7.2 2-colorability of certain hypergraphs . . . .
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8 The Pk -hitting set problem
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8.1 P4 -hitting set for general graphs . . . . . . . . . . . . . . . . . 92
9 Conclusion and further research
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Bibliography
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Index
111
Chapter 1
Introduction
In 1735, Leonhard Euler (1707–1783) solved a historically notable problem in
discrete mathematics: the seven bridges of Kőnigsberg, establishing the first
result in graph theory. The problem was to find a closed walk crossing exactly
once each bridge in the city of Kőnigsberg, in Prussia (now Kaliningrad, in
Russia). We can model this with a graph: each area is represented by a
vertex and each bridge by an edge (see Figure 1.1). The desired closed walk
is called a Eulerian cycle. Euler established a necessary condition for the
existence of a Eulerian cycle: the number of edges incident to every vertex
must be even. Actually this condition is also sufficient, provided the graph
is connected.
Figure 1.1: The seven bridges of Kőnigsberg and the corresponding graph.
Dénes Kőnig (1884–1944) was another forerunner of graph theory. In
1936, he wrote the first textbook on the field and proved a well-known theorem named after him: in every bipartite graph, the maximum number of
vertex-disjoint edges equals the minimum number of vertices meeting all the
edges, or more formally, the number of edges in a maximum matching equals
the number of vertices in a minimum vertex cover.
Another founding father of graph theory is Claude Berge (1926–2002).
He is well-known in particular for his contributions to perfect graphs. A
1
2
Chapter 1. Introduction
graph is perfect if the chromatic number of every induced subgraph equals
the size of the largest clique of this subgraph (the chromatic number is the
minimum number of colors to assign to vertices such that any two adjacent vertices have distinct colors, and a clique is a set of pairwise adjacent
vertices). Berge proposed two conjectures on perfect graphs. Berge’s first
conjecture stated that a graph is perfect if and only if its complement is
perfect. The complement G of a graph G is the graph on the same vertices
such that two vertices are adjacent in G if and only if they are not in G. The
second conjecture is a characterization of perfect graphs in terms of forbidden induced subgraphs: a graph G is perfect if and only if neither G nor G
contains an induced cycle Ck on k vertices, with k > 5 an odd number.
After Berge, (structural) graph theory emerged as an important discipline
of discrete mathematics. At the same time, major progress was achieved in
computer science, especially in computational theory, whose main focus is
to determine whether a computational problem can be solved efficiently in
standard models of computation.
The introduction of the class P goes back to Jack Edmonds (1934–)
and Alan Cobham (1927–). This class contains all decision problems that
can be solved by a polynomial-time algorithm. Edmonds and Cobham first
proposed polynomial-time solvability as a synonym for tractable.
Beyond P , the class N P contains all decision problems whose certificate
can be checked in polynomial-time. By definition, P is a subset of N P . An
important open question is whether this inclusion is strict, that is, whether
P 6= N P . Stephen Cook (1939–) showed that inside N P , there are problems
which are at least as difficult as all the problems in N P because every such
problem can be reduced to it. Richard Karp (1935–) brought the concept
of N P -completeness to the attention of a larger public with his famous list
of 21 N P -complete problems coming from different fields [83]. Among these
was the vertex cover problem: given a graph G and a positive integer k,
determine whether G has a vertex cover of size at most k.
The class of N P -complete problems includes the vertex cover problem
and the dominating set problem, which will play a major role in our investigations. A dominating set is a set S of vertices such that every vertex
not in S is adjacent to a vertex in S. The dominating set problem can be
formulated as follows: given a graph G and a positive integer k, determine
whether G has a dominating set of size at most k. Both problems are special
cases of the H -hitting set problem.
Given a graph G and a collection H of subgraphs of G, we define an
H -hitting set as a set of vertices of G meeting all subgraphs of H . For the
vertex cover problem, the collection H contains all the edges while, for the
dominating set problem, a subgraph in H is induced by a vertex and all its
neighbors. The H -hitting set problem consists in finding an H -hitting set
of minimum size.
Because it is widely believed that no polynomial-time algorithm exists for
3
N P -complete problems, the computer science community gave up on trying
to find polynomial-time algorithms for solving these problems exactly and,
instead, investigated heuristics. These are efficient algorithms designed to
find a “good” feasible solution to a given optimization problem. Of particular
interest are approximation algorithms, which are heuristics with a guarantee
on the quality of the solution.
A canonical example of an approximation algorithm is the following simple greedy procedure for the vertex cover problem: iteratively find an uncovered edge and add both endpoints to the vertex cover, until no uncovered
edge remains. Since the number of edges in any matching is a lower bound
on the size of every vertex cover, the resulting vertex cover is at most twice
as large as the optimal one. This is an approximation algorithm with a
performance ratio of 2.
This thesis involves two points of view on graph theory. The first one
is structural as illustrated by the contributions of Euler, Kőnig and Berge
described above. The second one is algorithmic. Both sides have their importance and are closely related: their interplay fosters the development of
graph theory. Indeed, structural results give tools to design algorithms, while
algorithmic problems motivate the study of structural questions.
A typical instance of such an interaction between structural and algorithmic results comes from the theory of graph minors. The concept of minor generalizes the concept of subgraph by allowing, besides the operations
of edge-deletion and vertex-deletion, that of edge-contraction. Kuratowski
proved the following theorem about planar graphs (defined as graphs that
can be drawn in the plane R2 without crossing internally edges): a graph is
planar if and only if none of its minors includes K5 or K3,3 .
Figure 1.2: K5 on the left and K3,3 on the right.
Kuratowski’s theorem can be generalized to graphs embedded in any
given surface of R3 . More generally, from 1983 to 2004, Neil Robertson and
Paul Seymour proved inter alia that every family of graphs that is closed
under minors can be characterized by a finite set of forbidden minors.
For instance, a minor-closed family is the set of graphs with treewidth
bounded by some fixed constant k. Intuitively, the treewidth measures how
close a given graph is to a tree. Treewidth is a very important graph invariant
in algorithmic graph theory. Robertson and Seymour showed that if the
4
Chapter 1. Introduction
treewidth of a graph is large, then it contains a somewhat large grid minor.
In this instance, structural graph theory gives tools to develop algorithms on
graphs. Typically, when a problem is easy to solve on trees, it is expected
to be easy to solve on bounded-treewidth graphs.
Before explaining how this interaction materializes in this thesis, we introduce the connectivity constraint, which will be present throughout our
work. We consider the connected version of the H -hitting set problem, in
which we require the H -hitting set to induce a connected subgraph. Clearly,
the minimum size of a connected H -hitting set is always at least that of an
H -hitting set. We define the price of connectivity as the following ratio:
price of connectivity =
minimum size of a connected H -hitting set
minimum size of an H -hitting set
The main goal of this thesis is to study the price of connectivity for the
vertex cover problem and the dominating set problem.
This is a natural structural question motivated, for instance, by the following two-phase algorithmic approach to the connected H -hitting set problem: first find an optimal solution to the H -hitting set problem and then
transform it into a connected H -hitting set without increasing its size too
much. Thus, in some sense, the structural results obtained in this thesis
are motivated by an algorithmic problem. The algorithmic origin of the
questions survives in our proof techniques.
As we will prove, computing the price of connectivity is complete for a
class which is above N P , thus not much can be said in general about graphs
with price of connectivity bounded by a given rational number r. However,
the situation is completely different once we consider restricted classes of
graphs, such as those defined by forbidden induced subgraphs. This explains
why this thesis focuses mainly on such hereditary classes of graphs.
Coming back to the interplay between the structural and algorithmic
points of view, we will also prove structural results that have algorithmic
consequences. We give a new characterization of Pk -free graphs (that is,
graphs without any induced path on k vertices) in terms of connected dominating sets. We will prove that our characterization yields a polynomial-time
algorithm for solving the 2-colorability problem in a restricted class of hypergraphs.
1.1 Outline
1.1
5
Outline
This thesis tackles three distinct topics:
1. the price of connectivity for the H -hitting set problem,
2. a characterization of the class of Pk -free graphs,
3. the Pk -hitting set problem.
We now give a brief overview of the content of each chapter.
Chapter 2 – Background
We give basic definitions used in the following chapters. On the complexity side, we briefly recall essential notions such as complexity classes, Lreductions and approximation algorithms, and we give elementary definitions
involving graphs and hypergraphs. We then consider three famous problems
involving hitting sets in graphs: the vertex cover problem, the dominating
set problem and the feedback vertex set problem.
Chapter 3 – The price of connectivity and other prices
This chapter is completely devoted to giving the context of our first topic.
We briefly recall related notions in mathematics and computer science such
as the competitive ratio, the price of anarchy and the price of stability. Then
we state some of the previously known results on the price of connectivity:
• for the vertex cover problem, initiated by Cardinal and Levy [41, 88],
• for the dominating set problem, based on our Master thesis [30],
• for the feedback vertex set problem, studied by Belmonte, van ’t Hof,
Kamiński and Paulusma [16,17], Grigoriev and Sitters [70], and Schweitzer and Schweitzer [112].
Finally, we summarize several other works comparing different variants
of the domination invariant.
Chapter 4 – The price of connectivity for vertex cover
We continue the study of the price of connectivity for the vertex cover problem, initiated by Cardinal and Levy [41,88]. We investigate both complexity
and structural aspects.
First, we consider the following problem: given a graph G and a constant
r, is the price of connectivity of G at most r? Regarding computational
complexity, we establish that this decision problem is essentially as hard as
computing both the connected vertex cover number and the vertex cover
6
Chapter 1. Introduction
number of this graph. More precisely, computing the price of connectivity is
a Θp2 -complete problem.
Secondly, we consider an analogous problem in which the price of connectivity for all induced subgraphs of G is bounded by a fixed constant r. This
problem is clearly different from the previous one, at least for small values
of r. Indeed, we find a characterization in terms of a finite list of forbidden
induced subgraphs when r ∈ [1, 3/2]. In order to find such results for other
constants r, we define a PoC-critical graph as one that appears in the list
of minimal forbidden induced subgraphs for some threshold. Towards this
goal, we define the restricted subclass of PoC-strongly-critical graphs. Every
PoC-critical chordal graphs is also PoC-strongly-critical. Moreover, we also
characterize this class by special trees.
Besides, we answer the following natural question: for which rational
number r is there a graph whose price of connectivity is exactly r?
Chapter 5 – The price of connectivity for domination
This chapter is split into two sections and extends the work started in our
Master thesis [30].
For the vertex cover problem, we prove that, given a graph G and a
constant r, deciding whether the price of connectivity of G is at most r is
also Θp2 -complete. For a fixed constant r, the following decision problem is
different from the previous one: given a graph G, is the price of connectivity
for every induced subgraph at most r? Similarly to the vertex cover problem,
for r ∈ [1, 3/2], we characterize the class of graphs with a ‘yes’-answer to the
previous problem. Among other results, we prove that for any (P6 , C6 )-free
graph, the difference between the connected domination number and the
domination number is at most 1.
We also introduce likewise the notion of PoC-critical graphs and PoCstrongly-critical graphs for the dominating set problem.
Finally, for all rational number r ∈ [1, 3), we construct a graph whose
price of connectivity is exactly r.
Chapter 6 – A characterization of Pk -free graphs
This chapter introduces our second topic and gives a structural application of
connected dominating sets. The class of Pk -free graphs can be characterized
in terms of connected dominating sets: being Pk -free is equivalent to the fact
that every connected induced subgraph admits a connected dominating set
which is either isomorphic to Ck or Pk−2 -free.
Moreover, we describe a polynomial-time algorithm finding such a connected dominating set. Our algorithm is oblivious to the minimum value k
such that the graph is Pk -free.
1.1 Outline
7
Chapter 7 – 2-colorability of hypergraphs
We give an application of our Pk -free graph characterization for the 2-colorability of hypergraphs: the 2-colorability problem can be solved in polynomial
time for hypergraphs with P7 -free incidence graph. This generalizes previous
results of van ’t Hof and Paulusma [122] who proved this for hypergraphs
with P6 -free incidence graph. Before giving the proof, we give a brief survey
of the 2-colorability problem in hypergraphs by giving a detailed account
of known sufficient conditions and by mentioning some related complexity
results.
Chapter 8 – The Pk -hitting set problem
Our third and last topic concerns the H -hitting set problem, where H is
the set of all paths on k vertices in a graph G. This problem is as hard as
the vertex cover problem, for finding exact solutions but also in the sense
of approximation algorithms. For k = 3, there exists a polynomial-time
2-approximation algorithm [120], inspired from the primal-dual method for
the feedback vertex set problem [12,15,46]. At a higher level, the connection
between the two problems is explained by the fact that graphs not containing
P3 as a subgraph are very restricted forests. Unfortunately, for k > 4, graphs
containing no Pk as a subgraph may have cycles. This makes it more difficult
to adapt the primal-dual algorithm to the Pk -hitting set problem for k > 4.
A k-approximation algorithm can trivially be obtained by taking all vertices in an inclusion-wise maximal packing of vertex-disjoint subgraphs each
isomorphic to Pk . However, nothing better than a k-approximation algorithm is known for the general problem in graphs when k > 4.
In that chapter, we develop a primal-dual 3-approximation algorithm for
the P4 -hitting set problem.
8
Chapter 1. Introduction
1.2
Contributions
• The results of Chapter 4 were obtained in collaboration with Jean
Cardinal, Samuel Fiorini and Oliver Schaudt. They were presented
at the 11th Cologne-Twente Workshop on Graphs and Combinatorial
Optimization [33] and also at GraphDay@Mons and Young Women in
Discrete Mathematics. The results have been published in Discrete
Mathematics & Theoretical Computer Science [32].
• The results of Chapter 5, except those in the last two subsections,
were obtained with Oliver Schaudt, and presented at the 12th CologneTwente Workshop on Graphs and Combinatorial Optimization [37]. Besides, they have been published in Discrete Applied Mathematics [36].
• The characterization of Pk -free graphs and its application to the 2colorability of hypergraphs (Chapter 6 and 7) were originally presented
at 40th International Workshop on Graph-Theoretic Concepts in Computer Science [35] (WG2014). The paper received a “best paper award”
in WG2014 and resulted in a publication in Algorithmica [34]. This is
also joint work with Oliver Schaudt.
• The work on the Pk -hitting set problem (Chapter 8) was done in collaboration with Jean Cardinal, Mathieu Chapelle, Samuel Fiorini and
Gwenaël Joret. It was presented at the 9th International colloquium
on graph theory and combinatorics [31].
Chapter 2
Background
In this section, we give a brief introduction of basic definitions and graphtheoretic concepts. Since all relevant definitions are listed in the index of this
thesis, any person familiar with graph theory can skip the current chapter
and proceed to Chapter 3. We use the notations of Diestel [50]. A good
introduction to the theory of computation is given by Sipser [114].
2.1
Computational complexity
In computational complexity theory, we distinguish two main classes of problems: decision problems and optimization problems. A decision problem is a
problem where the expected answer is “yes” or “no”, whereas an optimization
problem involves a set of feasible solutions, and the expected answer is one
whose value of the objective function is optimal. An optimization problem
is a minimization (resp. maximization) problem when its objective function
must be minimized (resp. maximized). Note that the value of the objective function for a feasible solution is its objective value. Besides, for each
optimization problem, there exists a corresponding decision problem where
the inputs are the input of the optimization problem and some constant.
Then the form of the corresponding decision problem is: “Does there exist a
feasible solution whose objective value is bounded by the constant?”.
For instance, the set cover problem is an optimization problem. An
instance of the set cover problem is an ordered pair (U, S), where U is called
the universe and S is a family of subsets of U whose union equals the universe.
A set cover , a feasible solution, is a subfamily of S whose union equals
the universe. The objective function is the size of the subfamily. The set
cover problem is a minimization problem. For some inputs (U, S) and some
constant k, we can consider the decision problem: “Does there exist a set
cover of U with at most k subsets?”.
Some problems can be solved efficiently by an algorithm. The computational complexity theory classifies problems into classes according to the
9
10
Chapter 2. Background
effectiveness of their resolution. We present three classes encountered in this
thesis.
2.1.1
Classes P and N P
The classes P and N P are two of the most fundamental complexity classes.
A decision problem is in P if it can be solved by a deterministic algorithm
in polynomial time, whereas the class N P contains all decision problems
solvable by a non-deterministic algorithm in polynomial time. In practice,
the former concerns efficiently solvable or tractable problems. However, the
latter ensures that checking the feasibility of a ‘yes’-instance can be done by
a deterministic algorithm in polynomial time. We note the set of decision
problems that can be solved by a deterministic algorithm in a running time
O(f (n)) by DT IM E(f (n)). Consequently,
P =
[
DT IM E(nk ).
k∈N
For instance, primality testing is in P [1], whereas the decision problem
for set cover is in N P .
A major unsolved problem in computer science is whether P 6= N P . If
P 6= N P , some N P problems would be harder to compute than to verify.
One significant advance on this question came by distinguishing certain
problems in N P : if there exists a polynomial-time algorithm solving any
of these problems, all problems in N P would be solvable in polynomial
time. We call these problems N P -complete. The set cover problem is N P complete.
An N P -complete problem is a problem in N P which is N P -hard : at
least as hard as the hardest problems in N P . Formally, a problem P1 is
N P -hard if for every problem P2 in N P , there exists a polynomial-time
reduction from P2 to P1 . A polynomial-time reduction from P2 to P1 is a
polynomial-time algorithm transforming an input x to problem P2 into an
input y to problem P1 such that x is a ‘yes’-instance of P2 if and only if
y is a ‘yes’-instance of P1 . We denote the reduction from problem P2 to
P1 by P2 6p P1 . In practice, proving the N P -completeness of a problem P
consists of proving that P ∈ N P and P 0 6p P for one N P -complete problem
P 0 , because of the transitivity of the polynomial-time reduction.
2.1.2
Class Θp2
To compare the difficulties of problems, we introduce the concept of an oracle
machine: this is an algorithm with a black box, the oracle, which is able
to solve certain decision problems in a single operation. In particular, an
N P -oracle solves decision problems from N P in a single operation.
2.2 Approximation algorithms
11
The class Θp2 , sometimes denoted by P NP[log] , is defined as the class of
decision problems solvable in polynomial time by a deterministic algorithm
that allows using O(log n) many queries to an N P -oracle, where n is the size
of the input. Clearly, N P is included in Θp2 .
A Θp2 -complete problem is in Θp2 and is Θp2 -hard. Similarly to an N P complete problem, a problem P is Θp2 -complete if P ∈ Θp2 and P 0 6p P for
one Θp2 -complete problem P 0 . Spakowski and Vogel [116] proved the Θp2 completeness of the following decision problem: “given two graphs G1 and
G2 , is τ (G1 ) 6 τ (G2 )?”, where τ (G) is the minimum number of vertices of G
meeting all the edges of G, i.e. the minimum number of vertices of a vertex
cover.
2.2
Approximation algorithms
Because of such intractability concerns, alternative methods were developed
for optimization problems: heuristics are algorithms designed for finding
an approximate solution when an exact solution is out of reach. Approximation algorithms are polynomial-time heuristics whose solution is a good
approximation of the optimal one(s). The guarantee of the quality of this
approximation is measured by the performance ratio. Given a feasible solution x provided by an approximation algorithm, the performance ratio α of a
minimization problem (resp. maximization problem) is an upper (resp. lower)
f (x)
bound on the ratio f (OP
T ) , i.e. f (x) 6 αf (OP T ) (resp. αf (OP T ) 6 f (x)),
where f is the objective function and OP T is an optimal solution. In this
case, we say that the optimization problem is α-approximable or that the
algorithm is an α-approximation.
For instance,
the set cover problem is H(n)-approximable [47], where
Pn
H(n) = k=1 1/k is the nth harmonic number. This is achieved by using the
greedy algorithm: iteratively choose a set that contains the largest number
of uncovered elements.
By analogy to polynomial-time reductions in the case of decision problems, an L-reduction compares two optimization problems P1 , P2 with objective functions f1 , f2 and is defined by a quadruple (g, h, β, γ) as follows:
• for every instance x of P1 , g computes in polynomial time an instance
g(x) of P2 ,
• for every feasible solution y to g(x), h computes a feasible solution h(y)
of x in polynomial time,
• for every instance x of P1 , f2 (OP Tg(x) ) 6 βf1 (OP Tx ), where OP Tx
(resp. OP Tg(x) ) is an optimal solution for the instance x (resp. g(x)).
• for every feasible solution y to g(x),
|f1 (OP Tx ) − f1 (h(y))| 6 γ|f2 (OP Tg(x) ) − f2 (y)|.
12
Chapter 2. Background
In this case, P1 is said to be L-reducible to P2 (denoted by P1 6L P2 ).
Note that the constants β and γ allow to preserve a good performance
ratio. For instance, if P1 6L P2 with β = γ = 1 and P2 is α-approximable,
then P1 is also α-approximable.
2.3
Graphs and hypergraphs
A graph is an ordered pair G = (V, E) where V and E are finite sets and E
is a set of subsets of V containing exactly two elements of V . We suppose
that V and E are disjoint. The elements of V are called vertices and those
of E edges. If uv ∈ E with vertices u and v, then u and v are called the
endpoints of the edge uv. When the vertex or edge set of a graph G are not
specified, we denote by V (G) its vertex set and E(G) its edge set. Given a
graph G, the number of vertices |V (G)| is its order .
Two edges e and f are adjacent if e ∩ f 6= ∅. Two vertices u and v are
adjacent if E contains the edge uv. In this case, we say that u and v are
neighbors. The neighborhood NG (v) (sometimes called the open neighborhood ) of a vertex v is the set of all its neighbors. If we add the vertex v
to this set, we obtain the closed neighborhood NG [v] of v in G. A private
neighbor of a vertex v with respect to a vertex set S is a vertex u ∈
/ S such
that NG (u) ∩ S = {v}. The degree of a vertex v, denoted by dG (v), is the
number of its neighbors. A degree-0 vertex is isolated whereas a degree-1
vertex is pendent . A graph whose vertices all have the same degree d is
d-regular . We say that G is cubic if it is 3-regular. The maximum degree
(resp. minimum degree) of a graph G, denoted by ∆(G) (resp. δ(G)), is the
maximum (resp. the minimum) degree of all vertices of G. We omit the
graph G from the previous notations, for instance N (v), N [v], d(v), if there
is no possible confusion.
A path in a graph G is a sequence v1 v2 v3 · · · vk of k distinct vertices with
vi vi+1 ∈ E for any 1 6 i < k. The length of the path v1 v2 v3 · · · vk is the
number of its edges (i.e. k − 1) and the path v1 v2 v3 · · · vk connects vertices v1
and vk . An induced path is a path v1 v2 v3 · · · vk such that vi is not adjacent
to vj for any j ∈
/ {i − 1, i + 1}. By abuse of notation, the graph formed
by a sequence v1 v2 v3 · · · vk is also called a path and is denoted by Pk . The
distance between two vertices u and v, denoted by d(u, v), is the minimum
length of a path connecting them. The distance between two vertex sets X
and Y is min{d(x, y)|x ∈ X, y ∈ Y }. The diameter of a graph is the largest
distance between any pair of vertices. A vertex is central (or a center ) in
G if its largest distance from any other vertex is minimum. This distance is
the radius of G.
We say that H is a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G).
Let X be a subset of V (G). Then the unique subgraph with vertex set X
and edge set {uv | uv ∈ E(G), u, v ∈ X} is the subgraph induced by X and
2.3 Graphs and hypergraphs
13
is denoted by G[X]. Moreover, an induced subgraph of G is a subgraph of
G which is induced by some set X ⊆ V (G). The complement of a graph G,
denoted by G, is the graph with V (G) = V (G) such that two vertices are
adjacent in G if and only if they are not adjacent in G. Of course, G = G
for every graph G. Let X be a subset of V (G). G − X denotes the subgraph
induced by V (G) \ X. In particular, when X is reduced to the vertex v, we
denote it by G − v. Let Y be a subset of E(G). The graph G − Y is the
graph such that V (G − Y ) = V (G) and E(G − Y ) = E(G) − Y , especially
if Y is reduced to the edge e, G − e is the graph obtained by removing from
G the edge e.
A graph G is connected if there exists a path connecting each pair of
vertices from G. Otherwise the graph G is disconnected . The connected
components of a graph are the inclusion-wise maximal connected subgraphs.
A set of vertices X is a cutset of G if the number of connected components
of G − X is different from that of G, while a vertex v is a cutvertex of G if
the set {v} is a cutset of G. The disjoint union of some graphs G1 , . . . , Gk
with disjoint vertex and edge sets is the graph G with V (G) = ∪ki=1 V (Gi )
and E(G) = ∪ki=1 E(Gi ) and is denoted by G1 + G2 + · · · + Gk . We denote
by kG the disjoint union of k copies of G.
A cycle is a path v1 v2 v3 · · · vk where v1 is adjacent to vk . The length of
the cycle v1 v2 v3 · · · vk is also its number of edges, i.e. k. An induced cycle
is a cycle C = v1 v2 v3 · · · vk such that removing any edge from C results in
an induced path. By abuse of notation, the graph formed by this sequence
v1 v2 v3 · · · vk is also called a cycle and is denoted by Ck . An acyclic graph,
which is one not containing any cycle as a subgraph, is called a forest. A
connected forest is called a tree and any subgraph of a tree is a subtree. The
internal vertices of a tree are those with degree at least two. A linear forest
is a forest where each connected component is a path. A spanning tree T of a
graph G is a subgraph of G which is a tree with V (T ) = V (G). A maximum
leaf spanning tree is a spanning tree with a maximum number of leaves.
Two graphs G and H are isomorphic if there is a bijection φ : V (G) →
V (H) such that uv is an edge of G if and only if φ(u)φ(v) is an edge of H, for
every u, v ∈ V (G). If two graphs G and H are isomorphic, we write G ∼
= H.
A (graph) parameter is a function f from the set of all graphs to the natural
numbers N. A graph parameter f is an invariant if f (G) = f (H) whenever
the graphs G and H are isomorphic.
A clique in a graph is a vertex subset X whose vertices are pairwise
adjacent. When the vertex set of a graph G is a clique, we say that G is
complete. The complete graph on n vertices is denoted by Kn .
An independent set of a graph G is a vertex set X such that no two of
its elements are adjacent. We call the independence number of a graph G
the maximum size of an independent set. It is denoted by α(G).
A graph G is k-partite if there is a partition of V (G) in k parts such that
14
Chapter 2. Background
each part is an independent set. When k = 2, we usually obtain bipartite
graphs. The complete bipartite graph, where one block is of size n and the
other of size m with all possible edges between those two blocks, is denoted
by Kn,m . The graph K1,3 is a claw while, for every k > 0, the graph K1,k is
a star .
A matching is a set of edges such that no two edges share an endpoint.
If M is a matching such that every vertex is incident to some edge of M ,
then M is called a perfect matching.
The graph G is said to be H-free if no induced subgraph of G is isomorphic to H. Furthermore, we say that G is (H1 , . . . , H` )-free if G is Hi -free
for every i ∈ {1, . . . , `}.
A huge number of graph classes have been studied in literature. Below,
we list a few of them:
1. A chordal graph, or triangulated graph, is a graph in which all induced
cycles have length 3.
2. A planar graph is a graph that can be embedded in the plane R2 , i.e. it
can be drawn in R2 in such a way that its edges are internally disjoint.
3. A split graph is a graph whose vertices can be partitioned into a clique
and an independent set.
4. A Moore graph is a d-regular
graph with diameter k whose number
Pk−1
of vertices is exactly 1 + d i=0 (d − 1)i . Notice that the number of
vertices
graph with diameter k is upper bounded by
P of any d-regular
i.
(d
−
1)
1 + d k−1
i=0
5. A cograph is a P4 -free graph.
6. A trivially perfect graph is a (P4 , C4 )-free graph.
A hypergraph H is an ordered pair (V, E) where V is a finite set, called
the ground set of vertices, and E is a finite set of subsets of V . The elements
of E are called hyperedges. The rank of a hypergraph H is the maximum size
of its hyperedges. If all hyperedges have the same size k, the hypergraph is
said to be k-uniform. A graph is a 2-uniform hypergraph. The degree dH (v)
of a vertex v is the number of hyperedges that contain it. H is k-regular if
every vertex has exactly degree k. The (vertex-hyperedge) incidence graph
of a hypergraph H = (V, E) is the bipartite graph G with vertex set V ∪ E
and edge set {ve | v ∈ V, e ∈ E, v ∈ e}. A hypergraph H = (V, E) is
connected if there is no bipartition A ∪ B of V such that for all e ∈ E,
either e ⊆ A or e ⊆ B. For a hypergraph H = (V, E), a k-coloring is a
map c : V → {1, . . . , k} such that every hyperedge of size at least 2 is not
monochromatic, i.e. every such hyperedge contains at least two vertices of
distinct colors.
2.4 H -hitting set problems
2.4
15
H -hitting set problems
Given a graph G and a collection of subgraphs of G, we define an H -hitting
set as a set X of vertices such that for every H ∈ H , V (H) ∩ X 6= ∅. An
H -hitting set is minimum if its size is minimum. The H -hitting number ,
denoted by τH (G), is the size of a minimum H -hitting set. For a connected
graph G, an H -hitting set of G inducing a connected subgraph is called a
connected H -hitting set. Assume that G is disconnected, i.e. G admits the
connected components C1 , . . . , Ck . Let s 6 k be the number of connected
components C such that τH (C) 6= 0. Then a connected H -hitting set of
G is an H -hitting set whose number of connected components is exactly
s. Also, a connected H -hitting set whose size is minimum is a minimum
connected H -hitting set. The size of such a set, denoted by τH ,c (G), is
called the connected H -hitting number . An (resp. connected) H -hitting set
is minimal if none of its proper subsets is an (resp. connected) H -hitting set.
The (resp. connected) H -hitting set problem consists of finding a minimum
(resp. connected) H -hitting set. In this thesis, we are interested in three
particular collections H whose corresponding H -hitting set problems are
the vertex cover problem, the dominating set problem and the feedback
vertex set problem.
The following table describes definitions and notations for these three
problems, given a graph G.
Problems
set
H
τH (G)
τH ,c (G)
(connected)
H -hitting
number
Vertex cover
problem
Dominating set
problem
Feedback vertex set
problem
vertex cover set
all edges
dominating set
all stars induced by
a closed neighborhood
feedback vertex set
all cycles
τ (G)
τc (G)
γ(G)
γc (G)
ρ(G)
ρc (G)
(connected)
vertex cover
number
(connected)
domination
number
(connected)
feedback vertex
number
X is a vertex cover (resp. feedback vertex set) of G if and only if V (G)\X
is an independent set (resp. a forest) in G. An alternative definition of
dominating sets is the following one: a dominating set of a graph G is a
vertex set D such that every vertex not in D has a neighbor in D, i.e.
∪v∈D NG [v] = V (G).
We now give a brief review of literature on these three problems.
16
2.4.1
Chapter 2. Background
The vertex cover problem
The vertex cover problem has been widely studied in literature and is one
of the 21 N P -complete problems identified by Karp [83] in 1972. Moreover,
Garey, Johnson and Stockmeyer [63] proved that it remains N P -complete
in cubic graphs. As is well known, the vertex cover problem admits a 2approximation algorithm, by the vertex set of an inclusion-wise maximal
matching or by the internal vertex set of a depth-first search tree [109],
and better performance ratio can be achieved: 2 − 2 ln(ln(n))
ln(n) (1 − o(1)) [74].
Dinur and Safra [52] proved that the vertex cover problem is N P -hard to
approximate to within a factor of 1.3606, unless P = N P . Moreover, it is
widely believed that this problem is hard to approximate to within 2−ε under
the Unique Games Conjecture [85], unless P = N P . For more explanations
about the Unique Games Conjecture, see [84].
For the connected version, Fernau and Manlove [58] showed that the connected problem is not approximable within a performance ratio of 1.3606 − δ
for any δ > 0, unless P = N P . Furthermore, Escoffier, Gourvès and Monnot [57] proved that this problem is polynomial in chordal graphs and is
5/3-approximable in any class of graphs where the vertex cover problem is
polynomially solvable, especially in bipartite graphs.
2.4.2
The dominating set problem
Like the vertex cover problem, the dominating set problem has been intensively studied in literature. The dominating set problem is N P -complete [62,
p. 190] by a reduction from the vertex cover problem. Moreover, Kann [79,
pp. 108–109] described a pair of polynomial-time L-reductions between the
dominating set problem and the set cover problem, which preserves the
performance ratio. In other words, if there exists a polynomial-time αapproximation algorithm for the dominating set problem, then the reduction
gives a polynomial-time α-approximation algorithm for the set cover problem and vice versa. Therefore, the dominating set problem is (1 + ln(n))approximable by the greedy algorithm [47]. Raz and Safra [104] showed that
no polynomial-time approximation algorithm can run within a ratio better
than c ln(n) for some c > 0 unless P = N P , for the set cover problem,
hence also for the dominating set problem. Recently, Alon, Moshkovitz and
Safra [4] proved a similar result with higher values of c, for instance when
c = 0.2267. In terms of exact algorithms, Fomin, Kratsch and Woeginger [60] designed exponential algorithms whose computational complexity is
in O(1.93782n ) whereas Grandoni [69] developed one with a better performance in O(1.8021n ). Later, Fomin, Grandoni and Kratsch [80, pp. 284-286]
deduced a branch & reduce algorithm in O(1.52626n ).
Among the applications of connected dominating sets, we mention the
routing of messages in mobile ad-hoc networks. Blum, Ding, Thaeler and
2.4 H -hitting set problems
17
Cheng [21] explained the usefulness of connected dominating sets in this
context. From a theoretical point of view, the connected dominating set
problem is equivalent to finding a maximum leaf spanning tree. Garey
and Johnson [62, pp. 206] explained that the last problem is N P -complete.
Therefore, it is the same for the connected dominating set problem. Guha
and Khuller [71] designed two approximation algorithms with ratio 4 +
2 ln(∆) and 3 + ln(∆) (where ∆ is the maximum degree of G) for the
connected dominating set problem, and they proved that there is no approximation algorithm with performance ratio ρH(∆) for ρ < 1 unless
N P ⊆ DT IM E(nO(ln ln(n)) ), where H is the Harmonic function. While
Ruan, Du, Jia, Wu, Li and Ko [108] developed an approximation algorithm
with performance ratio 2 + ln(∆), Du, Graham, Pardalos, Wan, Wu and
Zhao [54] showed that there exists an approximation algorithm with performance ratio a(1 + ln(∆ − 1)), for any a > 1.
2.4.3
The feedback vertex set problem
Karp [83], and more generally Lewis and Yannakakis [89], proved the N P completeness of the feedback vertex set problem. Moreover, approximation
algorithms [12, 15, 46] with performance ratio 2, for instance by the primaldual method, or exact exponential algorithm [59] in O(1.7548n ) were designed. However, Guruswami and Lee [73] proved recently the strong N P hardness of approximation result for a variant: under the Unique Games
Conjecture, for any integer k > 3 and ε > 0, it is hard to find a (k − ε)approximate solution to the problem of intersecting every cycle of length at
most k.
Surprisingly, the connected version has not been studied in literature until
recently. Belmonte, van ’t Hof, Kamiński and Paulusma [16,17] investigated
the connected feedback vertex set problem compared to the feedback vertex
set problem. We discuss their contributions in detail in Chapter 3. Grigoriev
and Sitters [70] studied also the connected feedback vertex set problem for
restricted classes of graphs. They proved the N P -hardness of this problem
for planar graphs with maximum degree 9 and, for any ε > 0, they designed
for planar graphs of minimum degree 3 an approximation algorithm with
performance ratio (1 + ε).
18
Chapter 2. Background
Chapter 3
The price of connectivity and
other prices
The price of connectivity, abbreviated by PoC, expresses the interdependence
of the connected version of a graph invariant and the original invariant.
Many authors studied other prices by comparing a graph invariant with
some variants of this invariant or by comparing various variants of one graph
invariant. This chapter retraces related works and is split into three sections.
The first one mentions famous prices in mathematics or computer science, in
different areas compared to graph theory. The second one is dedicated to the
price of connectivity, especially on the vertex cover problem, the dominating
set problem and the feedback vertex set problem. The last one investigates
other prices involving domination numbers.
3.1
Famous prices
Comparisons between related parameters of discrete structures are ubiquitous in mathematics and computer science. We present a brief survey of
some popular ones.
In computer science, alongside the performance ratio for approximation
algorithms, the competitive ratio deals with on-line algorithms in a theory
starting with the work of Sleator and Tarjan [115]. An on-line algorithm is
one that receives a sequence of requests and performs an immediate action
in response to each request. The novelty of their paper [115] lies in a new
measure of performance, the competitive ratio for on-line algorithms. The
competitive ratio of an algorithm is defined as the worst-case ratio between
its cost and that of a hypothetical offline algorithm which knows the entire sequence of requests in advance and chooses its actions optimally. An
algorithm is competitive if its competitive ratio is bounded. Competitive
algorithms are used to overcome uncertainties about the future, in the case
of on-line requests from a server. Many authors [7, 9, 10, 18, 25, 82, 86, 95]
19
20
Chapter 3. The price of connectivity and other prices
developed competitive algorithms and proved upper and lower bounds on
the competitive ratios achievable by on-line algorithms.
In applied mathematics, game theory [8,38,97,98] has been used to study
a wide variety of human and animal behaviors. The applications are manifold: modeling, economy, business, political science, biology, computer science, logic, philosophy, . . . In this context, many authors [2, 6, 43–45, 87, 90,
106, 107] established two famous notions: the price of anarchy and the price
of stability. A good introduction on the game theory is given by Nisan,
Roughgarden, Tardos and Vazirani [98].
First of all, the price of anarchy of a game is a concept that measures
how the efficiency of a game degrades due to selfish behavior of its players, in
other words, the ratio between the worst welfare function value of one of its
Nash equilibria and that of an optimal outcome. Notice that if the price of
anarchy is closer to 1, choosing an arbitrary Nash equilibrium as a solution is
relevant since the welfare function evaluated in any Nash equilibrium seems a
good approximation to the optimal value. Some authors [43–45, 87, 106, 107]
attempt to bound the price of anarchy in particular cases. Unfortunately, a
game with multiple Nash equilibria has a large price of anarchy even if only
one of its equilibria is highly inefficient.
Secondly, the price of stability is a measure of inefficiency designed to
differentiate between games in which all equilibria are inefficient and those
in which some equilibrium is inefficient. Formally, the price of stability
of a game is the ratio between the best welfare function value of one of
its Nash equilibria and that of an optimal outcome. Of course, in a game
with a unique equilibrium, its price of anarchy and price of stability are
identical. In general, the price of stability is relevant for games in which
there is some objective authority that can partly influence the players, and
can help them converge to a good Nash equilibrium. As the case of the price
of anarchy, bounding the price of stability is a challenge raised by several
authors [2, 6, 90]. Obviously, for a game with multiple equilibria, its price of
stability is at least as close to 1 as its price of anarchy, and it can be much
closer.
3.2
3.2.1
Price of connectivity
Vertex cover problem
The price of connectivity has been introduced by Cardinal and Levy [41,
88] for the vertex cover problem and is defined by the ratio between the
connected vertex cover number τc and the vertex cover number τ .
Let us first note that every vertex cover C of a graph G such that G[C]
has c connected components can be turned into a connected vertex cover
of G by adding at most c − 1 vertices. This directly yields the following
observation.
21
3.2 Price of connectivity
Observation 3.1. For every graph G it holds that τc (G) 6 2τ (G) − 1.
As an immediate consequence of Observation 3.1, the following inequality
holds for every graph G (with at least one edge):
1 6 τc (G)/τ (G) < 2.
(3.1)
Hence the price of connectivity for the vertex cover problem of any graph lies
in the interval [1, 2). Note that the upper bound in (3.1) is asymptotically
sharp in the class of paths Pk and in the class of cycles Ck on k vertices, in
the sense that
lim τc (Pk )/τ (Pk ) = 2 = lim τc (Ck )/τ (Ck ).
k→∞
k→∞
First of all, Cardinal and Levy [41, 88] showed that for the vertex cover
problem, the price of connectivity in dense graphs is bounded by a constant
depending on the graph density.
Theorem 3.2 (Cardinal, Levy [41, 88]). Let G be a graph with at least ε n2
edges. Then its price of connectivity for the vertex cover problem is at most
2
1+ε + O(1).
Moreover, they proved that this theorem is tight for the family of graphs
Gx,y with y−x a multiple of 3, defined as follows: Gx,y is the graph composed
of a clique of size x and (y − x)/3 paths on 3 vertices, all endpoints of which
are totally joined to the clique. Figure 3.1 is an illustration of G4,10 and
G6,18 .
Figure 3.1: On the left, the graph G4,10 and on the right, the graph G6,18 .
22
3.2.2
Chapter 3. The price of connectivity and other prices
Dominating set problem
First of all, Duchet and Meyniel [55] observed that for every graph G it
holds that γc (G) 6 3γ(G) − 2. As an immediate consequence, every graph
G satisfies
1 6 γc (G)/γ(G) < 3,
(3.2)
that is, the price of connectivity for dominating set problem of a graph G,
γc (G)/γ(G), is strictly bounded by 3.
Observation 3.3. It holds that
lim γc (Pk )/γ(Pk ) = 3 = lim γc (Ck )/γ(Ck ).
k→∞
k→∞
(3.3)
In particular, the upper bound (3.2) is asymptotically sharp in the class of
paths and in the class of cycles.
Moreover, Zverovich [126] found a characterization of a particular class
of graphs. Each graph of this particular class has the price of connectivity
for the dominating set problem equal to 1, like all induced subgraphs. The
following theorem is our starting point in Chapter 5.
Theorem 3.4 (Zverovich [126]). The following assertions are equivalent for
every graph G.
(i) For every induced subgraph H of G it holds that γc (H) = γ(H).
(ii) G is (P5 , C5 )-free.
Even though the class of (P5 , C5 )-free graphs is recognized in polynomial
time, the dominating set problem restricted to this class is N P -complete,
as proved by Bertossi [19] and by Corneil and Perl [48]. By the previous
theorem, the connected dominating set problem is also N P -complete.
During her Master thesis, Camby [30] studied the price of connectivity
for the dominating set problem. She proved that for an arbitrary constant
δ, there exists a sequence of graphs with minimum degree δ and price of
connectivity approaching 3 for the dominating set problem. This infinite
family of graphs G∗n,δ is defined as follows. Consider n disjoint Kδ+1 and
n − 1 disjoint Kδ−1 . We place alternately these cliques along a path and we
add edges as follows. In each clique Kδ+1 , choose two distinct vertices, say
u, v, and add all possible edges between u and the previous clique Kδ−1 , and
also between v and the following clique Kδ−1 . Notice that δ(G∗n,δ ) = δ. The
graph G∗4,3 is illustrated by Figure 3.2.
Moreover, for the dominating set problem, the maximum value of the
price of connectivity for graphs with minimum degree δ is decreasing with
respect to δ.
3.2 Price of connectivity
23
Figure 3.2: Graph G∗4,3 .
Theorem 3.5 (Camby [30]). Let n > 4. Consider the function f defined by
γc (G) δ(G) = δ, |V | = n .
f : {1, . . . , n − 1} → R : δ 7→ max
γ(G) Then f is decreasing.
As a result, for the dominating set problem, the maximum value of the
price of connectivity for graphs of order n is attained by a graph with minimum degree 1.
Besides, Camby found upper bounds when the minimum degree is proportional to the order of the graph.
Theorem 3.6 (Camby [30]). Let n > 4 and G be a graph of order n with
minimum degree at least n/2. Then γc (G)/γ(G) < 2.
2
Moreover, if δ(G) = n/2 > 3 then γc (G)/γ(G) 6 2 − n−2
.
The last bound may be not tight. However, Camby constructed by induction a graph G of order 2δ(G) whose price of connectivity for the dominating
set problem is equal to 3/2. The basic case is illustrated by Figure 3.3. We
conjecture that for any graph G with δ(G) = n/2, the price of connectivity
for the dominating set problem is bounded by 3/2.
Figure 3.3: Graph of order 8 with minimum degree 4 and price of connectivity
for the dominating set problem 3/2.
Theorem 3.7 (Camby [30]). Let n > 4 and δ ∈ N0 . Let G be a graph of
order n with minimum degree δ. If nδ > 3 then
γc (G)
2
63− n
.
γ(G)
dδe − 1
24
Chapter 3. The price of connectivity and other prices
Furthermore, Camby studied the price of connectivity for the dominating
set problem in some particular classes of graphs.
Theorem 3.8 (Camby [30]). Let G be a graph.
• If G is a cograph, i.e. a P4 -free graph, then γc (G)/γ(G) = 1.
• If the diameter of G is 2 then γc (G)/γ(G) < 2.
However, we conjecture that the upper bound on the price of connectivity
for the dominating set problem for graphs of diameter 2 is 3/2. Moreover,
there exists a sequence of graphs of order n + 3 with diameter 2 whose price
of connectivity for the dominating set problem is exactly 3/2. An instance of
these graphs, obtained by subdividing an arbitrary edge of K2,n , is depicted
in Figure 3.4.
Figure 3.4: Graph obtained by subdividing one edge from K2,10 .
3.2.3
Feedback vertex set problem
Belmonte, van ’t Hof, Kamiński and Paulusma [16, 17] studied the price of
connectivity for feedback vertex set problem, defined as the ratio between
the connected feedback vertex number ρc and the feedback vertex number ρ.
In general, the price of connectivity can be arbitrarily large for the feedback vertex set problem, for instance for butterflies. A butterfly is a graph
consisting of two disjoint cycles on i and j vertices that are connected to
each other by a path of length k. We denote it by Bi,j,k . The butterfly
B4,8,2 is illustrated by Figure 3.5. Notice that for the vertex cover problem
and the dominating set problem, the price of connectivity for any graph is
bounded by a fixed constant. At the opposite, the price of connectivity for
the feedback vertex set problem can be arbitrarily large since its exact value
for the butterfly Bi,j,k is (k + 2)/2.
Observe that butterflies are planar. However, Grigoriev and Sitters [70]
showed that the price of connectivity for feedback vertex set problem is at
most 11 for planar graphs of minimum degree at least 3. Later, Schweitzer
and Schweitzer [112] improved this upper bound down to 5, which is tight.
Belmonte, van ’t Hof, Kamiński and Paulusma, [17] established different
links between the connected feedback vertex number and the feedback vertex
number for the class of H-free graphs.
3.2 Price of connectivity
25
Figure 3.5: Butterfly B4,8,2 .
Theorem 3.9 (Belmonte et al. [17]). Let H be a graph.
• There is a constant dH such that ρc (G) 6 dH ρ(G) for every connected
H-free graph G if and only if H is a linear forest.
• There is a constant cH such that ρc (G) 6 ρ(G)+cH for every connected
H-free graph G if and only if H is an induced subgraph of P5 + sP1 or
sP3 for some integer s.
• ρc (G) = ρ(G) for every connected H-free graph G if and only if H is
an induced subgraph of P3 .
• For every constant eH , there is a H-free graph G with ρc (G) > eH ρ(G)
if and only if H contains a cycle as a subgraph or a vertex of degree at
least 3.
Afterwards, Belmonte et al. [16] generalized their results when the list
of forbidden induced subgraphs is finite. Let us introduce the following
definition to explain the following result.
Let i, j > 3 be two integers, let H be a finite family of graphs, and
let N = 1 + 2 maxH∈H |V (H)|. The family H covers the pair (i, j) if H
contains an induced subgraph of the butterfly Bi,j,N . A graph H covers the
pair (i, j) if the family {H} covers (i, j).
The following theorem states that the price of connectivity for feedback
vertex set problem in the class of H -free graphs is bounded by a constant
fH if and only if the forbidden induced subgraphs in H prevent arbitrarily
large butterflies from appearing as induced subgraphs.
Theorem 3.10 (Belmonte et al. [16]). Let H be a finite family of graphs.
Then the price of connectivity for feedback vertex set problem in H -free
graphs is upper bounded by a constant fH if and only if H covers the pair
(i, j) for every i, j > 3.
By Theorem 3.9, we know that covering all pair (i, j) for i, j > 3 is
equivalent to being a linear forest, completing the case when |H | = 1. Then,
they obtained a similar characterization when |H | = 2 in the same taste
than Theorem 3.10. First, we define a lollipop Lm,n as the graph obtained
by joining a complete graph Km to a path Pn with a bridge. A lollipop is
26
Chapter 3. The price of connectivity and other prices
simple when the complete graph has only 3 vertices and is denoted by Ln .
A simple lollipop is eaten if it is the subgraph of Ln obtained by removing
the edge to have no degree-2 vertex in the clique. An eaten simple lollipop
is denoted by In . These graphs are illustrated in Figure 3.6. Notice that L0
is isomorphic to C3 , the cycle on 3 vertices.
Figure 3.6: On the left, the simple lollipop L6 and on the right, the eaten
simple lollipop I3 .
Theorem 3.11 (Belmonte et al. [16]). Let H1 and H2 be two graphs, and
let H = {H1 , H2 }. Then the price of connectivity for feedback vertex set
problem in H -free graphs is upper bounded by a constant gH if and only if
there exist integers n > 0 and r > 1 such that one of the following conditions
holds:
- H1 or H2 is a linear forest,
- H1 and H2 are induced subgraphs of Ln and 2Ir , respectively,
- H1 and H2 are induced subgraphs of 2Ln and Ir , respectively,
where 2G is the disjoint union of two copies of G.
Clearly, Theorem 3.11 generalizes Theorem 3.9, as the class of H-free
graphs is equivalent to the class of {H, H}-free graphs. They pointed out
that any graph H that is an induced subgraph of both Ln for some n > 0
and of 2Ir for some r > 1 is a linear forest.
3.3
Other prices
We can also examine other pair of invariants in graph theory. For instance,
Fulman [61] and Zverovich [127] investigated the ratio between the independence number and the upper domination number , which is the maximum size
of an inclusion-wise minimal dominating set. However, we stay focused on
our topic: pair of parameters involving the same invariant.
Indeed, we study different variants of domination. A survey on the ratios
of some important domination invariants is described by Blidia, Chellali
and Favaron [20] for general graphs and for claw-free graphs while Chellali,
Favaron, Haynes and Raber [42] investigated the class of trees.
27
3.3 Other prices
3.3.1
Paired-domination versus total domination
Haynes and Slater [75] gave the following relation between the total domination number γt and the paired-domination number γp . A total dominating
set X is a vertex set such that every vertex, even in X, has a neighbor in X
whereas a paired-dominating set is a dominating set whose induced subgraph
has a perfect matching. Moreover, the total domination number , resp. the
paired-domination number , is the minimum size of such a dominating set.
Theorem 3.12 (Haynes, Slater [75]). Let G be a graph with minimum degree
at least 1. Then
γp (G) 6 2γt (G) − 2.
Therefore, for any graph G with minimum degree at least 1,
γp (G)
< 2.
γt (G)
This bound is asymptotically sharp in the sense that
γp (cr(K1,r ))
2
=2−
−→ 2
γt (cr(K1,r ))
r + 1 r→+∞
where the corona cr(G) of a graph G is the graph obtained from G by
attaching a pendent vertex to every vertex. Dorbec, Henning and McCoy [53] proved similar results for the ratio Γp /Γt between the upper paireddomination number and the upper total domination number, where the upper
paired-domination number Γp is the maximum size of an inclusion-wise minimal paired-dominating set while the upper total domination number Γt is
the maximum size of a minimal total dominating set.
Restricted to some classes of graphs, the upper bound on the ratio γp /γt
could be smaller. Indeed, Brigham and Dutton [29] showed that for K1,3 free graphs with minimum degree at least 1, the bound γp /γt 6 4/3 holds,
whereas Schaudt [111] generalized the upper bound to the class of K1,r -free
graphs.
Theorem 3.13 (Schaudt [111]). Let G be a K1,r -free graph with minimum
degree at least 1, for some r > 3. Then
γp (G)
2
62−
γt (G)
r
and this bound is sharp for each r > 3. Moreover,
Γp (G)
2
62− .
Γt (G)
r
28
Chapter 3. The price of connectivity and other prices
Unfortunately, the second upper bound is possibly not sharp. However,
Schaudt enhanced the last theorem by finding a class of graphs where both
upper bounds are tight. It is the class of (C5 , Hr )-free graphs, for some
r > 3, where Hr is the graph obtained from K1,r by subdividing each edge
exactly once. As a consequence of Theorem 3.13, Schaudt also obtained the
following corollary.
Corollary 3.14 (Schaudt [111]). Let G be a graph with minimum degree at
least 1 and maximum degree ∆,
γp (G)
2
62−
γt (G)
∆+1
and this bound is sharp. Moreover,
Γp (G)
2
62−
.
Γt (G)
∆+1
Besides, Schaudt established the following characterization on the ratio
γp /Γt .
Theorem 3.15 (Schaudt [111]). Let G be a graph with minimum degree at
least 1. The following assertions are equivalent.
(i) γp (H) 6 Γt (H) for any induced subgraph H of G with minimum degree
at least 1.
(ii) G is (C5 , cr(K3 ), cr(P3 ))-free (see Figure 3.7).
Figure 3.7: C5 , cr(K3 ) and cr(P3 ).
Furthermore, similar to Theorem 3.13, Schaudt [111] proposed the following theorem for the ratio γp /Γt .
Theorem 3.16 (Schaudt [111]). Let G be a cr(K1,r )-free graph with minimum degree at least 1, for some r > 3. Then
γp (G)
2
62−
Γt (G)
r
and this bound is sharp for each r > 3.
29
3.3 Other prices
In particular, Schaudt found the following corollary.
Corollary 3.17 (Schaudt [111]). Let G be a connected graph with minimum
degree at least 1 and maximum degree ∆ > 2 that is not isomorphic to C5 ,
γp (G)
2
62−
Γt (G)
∆
and this bound is sharp.
3.3.2
Connected domination versus total domination
A graph is called perfect if the chromatic number of every induced subgraph
equals the size of the largest clique of this subgraph, where the chromatic
number is the minimum value k such that the graph admits a k-coloring.
More generally, we can define perfection with other parameters. For instance, we can consider the domination number and the minimum size of
a dominating set whose induced subgraph is independent. Zverovich and
Zverovich [128] gave a minimal forbidden subgraph characterization of such
domination perfect graphs. We recall that Zverovich [126] gave a characterization of perfect graphs for the connected domination number and the
domination number, which said that γ = γc for any induced subgraph is
equivalent to being (C5 , P5 )-free. We define the clique-domination number
γcl as the minimum size of a dominating set whose induced subgraph is a
clique. Goddard and Henning [64] extended Zverovich’s result to total domination and clique-domination.
Theorem 3.18 (Goddard, Henning [64]). Let G be a graph. The following
assertions are equivalent.
(i) Every connected induced subgraph has a dominating clique.
(ii) γ(H) = γt (H) for any connected induced subgraph H with γ(H) = 2.
(iii) γ(H) = γt (H) for any connected induced subgraph H with γ(H) > 2.
(iv) γ(H) = γcl (H) for any connected induced subgraph H with γ(H) > 2.
(v) γ(H) = γc (H) for any connected induced subgraph H with γ(H) > 2.
(vi) G is (C5 , P5 )-free.
Recently, Schaudt [110] studied the interdependence between the connected domination number and the total domination number, as explained
in the following theorem. In what follows, we define a connected graph as
non-trivial if it is not an isolated vertex.
Theorem 3.19 (Schaudt [110]). Let G be a graph. The following assertions
are equivalent.
30
Chapter 3. The price of connectivity and other prices
(i) For any non-trivial connected induced subgraph H of G, γc (H) 6 γt (H).
(ii) For any non-trivial connected induced subgraph H of G, γc (H) 6 Γt (H).
(iii) For any non-trivial connected induced subgraph H of G, γc (H) 6 γp (H).
(iv) For any non-trivial connected induced subgraph H of G, γc (H) 6 Γp (H).
(v) G is (P7 , C7 , F1 , F2 )-free (see Figure 3.8).
(vi) Any connected induced subgraph H of G has a connected dominating
subgraph X which is (P5 , G1 , G2 )-free (see Figure 3.9).
Figure 3.8: Graphs P7 , C7 , F1 and F2 .
Figure 3.9: Graphs P5 , G1 and G2 .
Observe that any connected graph G with γc (G) > 2 fulfills γc (G) >
γt (G). Accordingly, any (P7 , C7 , F1 , F2 )-free graph G with γc (G) > 2 satisfies
γc (G) = γt (G). Notice also that any connected split graph is in particular
(P7 , C7 , F1 , F2 )-free. Moreover, Bertossi [19] proved that the dominating set
problem in split graphs is N P -complete while Theorem 3.18 implies that
for any non-trivial connected (P5 , C5 )-free graph G, γ(G) = γc (G) = γt (G),
provided γ(G) > 2. Thus, the connected dominating set problem and the
total dominating set problem remain N P -complete even when restricted to
split graphs, or more generally in the class of connected (P7 , C7 , F1 , F2 )-free
graphs.
Chapter 4
The price of connectivity for
vertex cover
We study the interdependence of the vertex cover number, τ , and the connected vertex cover number, τc , both from a complexity-theoretic point of
view and in some hereditary classes of graphs. This chapter is essentially
based on a paper published in Discrete Mathematics & Theoretical Computer Science [32]. Let us first recall the basic observation.
Observation 4.1. For every graph G with at least one edge, it holds that
1 6 τc (G)/τ (G) < 2.
We define the price of connectivity (PoC) of a graph G as the ratio
τc (G)/τ (G), as Cardinal and Levy [41, 88] did.
This chapter is split into two parts. In the first part, we consider the
computational complexity of the problem of deciding whether the price of
connectivity of a graph given as input is bounded by a certain constant r.
We show the completeness of this problem with respect to a well-defined
complexity class in the polynomial hierarchy. In the second part, we investigate graph classes in which the price of connectivity of every induced
subgraph is bounded by a constant r with r ∈ [1, 2). Those classes will be
defined by forbidden induced subgraphs. The forbidden subgraph characterizations directly yield polynomial-time algorithms for recognizing graphs
in those classes. Many results in this area concern graph classes defined by
forbidden induced subgraphs. This line of research stems from the classical
theory of perfect graphs, for which the clique number and the chromatic
number are equal in every induced subgraph [67]. Moreover, we investigate
possible values of the price of connectivity for the vertex cover problem.
31
32
4.1
Chapter 4. The price of connectivity for vertex cover
Computational Complexity
The class ΘP2 = P NP[log] is defined as the class of decision problems solvable
in polynomial time by a deterministic Turing machine that allows using
O(log n) many queries to an N P -oracle, where n is the size of the input.
Theorem 4.2. Let 1 < r < 2 be a fixed rational number. Given a connected
graph G, the problem of deciding whether τc (G)/τ (G) 6 r is Θp2 -complete.
It is easy to see that the above decision problem belongs to Θp2 , since
both τ and τc can be computed using logarithmically many queries to an
N P -oracle by binary search. Thus, Theorem 4.2 is a negative result: loosely
speaking, it tells us that deciding whether the price of connectivity is bounded
by a certain constant is as hard as computing both τ and τc explicitely. And
this remains true even if the constant is not part of the input.
Our reduction stems from the decision problem whether for two given
graphs G and H it holds that τ (G) ≥ τ (H), which is known to be Θp2 complete due to Spakowski and Vogel [116]. It uses a gadgetry that allows
us to compare τ and τc on a single graph.
However, the problem of deciding of the price of connectivity is solved
in polynomial time for certain class of graphs, for instance in the class of
chordal graphs. Indeed, Escoffier, Gourvès and Monnot [57] proved that the
connected vertex cover decision problem is polynomial in this class of graphs.
Moreover, Rose, Lueker and Tarjan [105] show that an algorithm known as
lexicographic breadth-first search solves the vertex cover decision problem
in linear time by finding a perfect elimination ordering. Based on this, they
found an efficient recognition algorithm for chordal graphs.
Unfortunately, it seems that our proof does not apply to the case r = 1.
We leave it as an open problem to determine the computational complexity
of deciding whether τc (G)/τ (G) = 1 for a given graph G.
We now proceed to prove Theorem 4.2.
Lemma 4.3. Given a connected graph G with n vertices, one can construct
in linear time a graph G0 such that τ (G0 ) = n + τ (G) and τc (G0 ) = 2n.
Proof. With each vertex vS∈ V (G), associate three vertices v, v 0 , v 00 in V (G0 ),
and let E(G0 ) := E(G) ∪ v∈V (G) {vv 0 , v 0 v 00 }. A minimum vertex cover of G0
is the union of a minimum vertex cover of G with all vertices of the form
v 0 . On the other hand, a minimum connected vertex cover of G0 contains all
vertices v, v 0 .
Lemma 4.4. Given a graph G with n vertices and m edges, one can construct
in linear time a graph G0 such that τ (G0 ) = n + m + 1 and τc (G0 ) = n + m +
1 + τ (G).
33
4.1 Computational Complexity
Proof. For each edge e = uv ∈ E(G), define two vertices e, e0 of V (G0 ). For
each vertex v ∈ V (G), define three vertices v, v 0 , v 00 of V (G0 ). Finally, add
two vertices w, w0 to V (G0 ). The set of edges E(G0 ) is defined as follows.
For each edge e = uv ∈ E(G), the vertices e and e0 of V (G0 ) are adjacent,
and vertex e is adjacent to vertices u00 and v 00 . Similarly, for each vertex
v ∈ V (G), vertices v and v 0 of V (G0 ) are adjacent, and v is adjacent to
both v 00 and w. Finally, ww0 ∈ E(G0 ). The construction is illustrated in
Figure 4.1.
w0
u0
v0
w
e
u
v 00
u00
v
e0
Figure 4.1: Representation of an edge e = uv in the construction of G0 in
Lemma 4.4.
Since for each edge e ∈ E(G), the corresponding vertex e ∈ V (G0 ) is
adjacent to the degree-1 vertex e0 , it can be considered, without loss of
generality, to be part of any minimum vertex cover of G0 . The same remark
holds for vertices v ∈ V (G), and for the unique vertex w. Now the union
C ⊂ V (G0 ) of those vertices is a vertex cover of G0 , hence we have τ (G0 ) =
n + m + 1.
We now have to compute τc (G0 ). The previous vertex cover C is not
connected, as G0 [C] has exactly m + 1 connected components: one for each
edge of G, and one induced by w and the vertices v ∈ V (G). To make it
connected, we need to augment C with the fewest possible additional vertices
of the form v 00 for v ∈ V (G). Every such vertex v 00 will link the component
containing v to every vertex e ∈ E(G) of G0 such that v ∈ e. Hence the
minimum number of additional vertices to add to C is exactly the size τ (G)
of a minimum vertex cover of G. Thus τc (G0 ) = n + m + 1 + τ (G), as
claimed.
Proof of Theorem 4.2. Let r = r1 /r2 be a fixed rational number with 1 <
r < 2. It is clear that the problem is in Θp2 , so we proceed to the Θp2 hardness. Let G and H be two graphs. We reduce from the Θp2 -complete
problem of deciding, whether τ (G) ≥ τ (H) for two given graphs G and H
(see Spakowski and Vogel [116]).
We can assume that G and H are both connected. Otherwise, we choose a
vertex from each connected component of G (resp. H), add two new vertices
w and w0 , and put an edge from w to all chosen vertices and to w0 . Let G0
34
Chapter 4. The price of connectivity for vertex cover
(resp. H 0 ) be the graph obtained from G (resp. H) by this procedure. It is
clear that τ (G0 ) = τ (G) + 1 and τ (H 0 ) = τ (H) + 1. Hence, τ (G) ≥ τ (H)
if and only if τ (G0 ) ≥ τ (H 0 ). So we may assume that both G and H are
connected.
The reduction consists of the following five steps.
Step 1. Let v be any vertex of G. Starting with r2 disjoint copies of G,
we connect all r2 copies of v to a new vertex w. We then attach a pendent
vertex w0 to w. We denote the graph obtained by Gr2 . Let nG = |V (G)|.
Clearly, τ (Gr2 ) = r2 τ (G) + 1 and |V (Gr2 )| = r2 nG + 2.
Similarly we construct Hr1 from H. Let nH = |V (H)| and mH = |E(H)|.
Clearly, τ (Hr1 ) = r1 τ (H) + 1, |V (Hr1 )| = r1 nH + 2, and |E(Hr1 )| = r1 mH +
r1 + 1.
Step 2. We apply Lemma 4.3 to Gr2 to get G0r2 . We obtain
τ (G0r2 ) = |V (Gr2 )| + τ (Gr2 )
= r2 τ (G) + r2 nG + 3,
τc (G0r2 )
= 2|V (Gr2 )|
= 2r2 nG + 4.
We apply Lemma 4.4 to Hr1 to get Hr0 1 , and obtain
τ (Hr0 1 ) = |V (Hr1 )| + |E(Hr1 )| + 1
= r1 (nH + mH + 1) + 4,
τc (Hr0 1 )
= τ (Hr1 ) + |V (Hr1 )| + |E(Hr1 )| + 1
= r1 τ (H) + r1 (nH + mH + 1) + 5.
Step 3. We construct a new graph U by taking the disjoint union of G0r2
and Hr0 1 , and adding an edge uv such that u ∈ V (G0r2 ), v ∈ V (Hr0 1 ), and
both u and v are adjacent to a degree-1 vertex in G0r2 and Hr0 1 , respectively
(such vertices always exist).
By construction of U ,
τc (U ) = τc (G0r2 ) + τc (Hr0 1 )
= r1 τ (H) + r1 (nH + mH + 1) + 2r2 nG + 9,
τ (U ) = τ (G0r2 ) + τ (Hr0 1 )
= r2 τ (G) + r1 (nH + mH + 1) + r2 nG + 7.
Step 4. Let ϕ1 = 2r2 nG +r1 (nH +mH +1)+9 and ϕ2 = r2 nG +r1 (nH +
mH + 1) + 7. In this step, we determine two non-negative integers a and b
such that
a + 2b + ϕ1
= r.
(4.1)
a + b + ϕ2
Let p = max{|ϕ2 − ϕ1 |, |ϕ1 − 2ϕ2 |} > 0. Let a = p(2r2 − r1 ) + ϕ1 − 2ϕ2
and b = p(r1 − r2 ) + ϕ2 − ϕ1 . Since 1 < r < 2, note that 2r2 − r1 > 1 and
35
4.2 Structural results
r1 − r2 > 1. Therefore, a and b are non-negative integers. Now, we show
that a and b satisfy (4.1). The numerator in (4.1) becomes a + 2b + ϕ1 = pr1
whereas the denominator is a + b + ϕ2 = pr2 . Hence, the ratio equals to r.
This proves our claim.
Step 5. We now construct a graph U 0 from U as follows. Let v be a
vertex in U of degree 1 (such a vertex is always present). Let P 1 be the
graph obtained from the chordless path with vertex set {u1 , u2 , . . . , ua } by
attaching a pendent vertex to every member of {u1 , u2 , . . . , ua }. Let P 2 be
the graph obtained from the chordless path with vertex set {v1 , v2 , . . . , v2b }
by attaching a pendent vertex to every member of {v2 , v4 , . . . , v2b }. Let U 0
be the graph obtained from the disjoint union of U , P 1 , and P 2 by putting
an edge from v to u1 and to v1 . Since a, b ∈ O(ϕ1 + ϕ2 ), the above procedure
can be done in linear time in the size of the graph U .
By the construction of U 0 , we obtain
τc (U 0 ) = τc (U ) + a + 2b
= r1 τ (H) + a + 2b + ϕ1 ,
0
τ (U ) = τ (U ) + a + b
= r2 τ (G) + a + b + ϕ2 .
Recall that r = r1 /r2 . By (4.1), there is some non-negative integer c such
that a + 2b + ϕ1 = r1 c and a + b + ϕ2 = r2 c. Hence,
τc (U 0 )
r1 τ (H) + a + 2b + ϕ1
r1 τ (H) + r1 c
τ (H) + c
=
=
=r
.
0
τ (U )
r2 τ (G) + a + b + ϕ2
r2 τ (G) + r2 c
τ (G) + c
Thus, τc (U 0 )/τ (U 0 ) ≤ r if and only if τ (H) ≤ τ (G). This completes the
proof.
4.2
Structural results
Before establishing further results, we prove the following useful structural
lemma.
Lemma 4.5. Let G be a connected graph and let C be a vertex cover of G.
If (A, B) is a bipartition of the connected components of C with A, B 6= ∅,
there exists A ∈ A and B ∈ B such that the distance between A and B is
exactly 2.
Proof. Let (A, B) be a bipartition of the connected components of C. Since
C has a finite number of connected components, there exist A ∈ A and B ∈ B
such that the distance between them is minimum. Now we show that this
distance is 2. Otherwise, let x1 x2 · · · xn be a shortest path between A and B
with x1 ∈ A and xn ∈ B, where n > 4. In this case, no xi , i = 2, . . . , n − 1,
belongs to C. Otherwise, B is not one nearest component of B from A or
A is not one nearest component of A from B. Thus, the edge x2 x3 is not
covered by C, a contradiction with the definition of vertex cover.
36
Chapter 4. The price of connectivity for vertex cover
4.2.1
PoC-Perfect Graphs
As Theorem 4.2 shows, the class of connected graphs where τc (G)/τ (G) 6 r
holds (for any fixed rational r ∈ (1, 2)) is Θp2 -complete to recognize. However,
if we restrict our attention to hereditary graph classes, we are able to derive
the following results. Note that our characterizations yield polynomial-time
recognition algorithms, since the list of forbidden induced subgraphs is finite
in each case.
We first consider the hereditary class of graphs G for which τc (G) = τ (G),
referred to as PoC-Perfect graphs.
Theorem 4.6. The following assertions are equivalent for every graph G:
(i) For every induced subgraph H of G it holds that τc (H) = τ (H).
(ii) G is (P5 , C5 , C4 )-free.
(iii) G is chordal and P5 -free.
Proof. The class of graphs that are chordal and do not contain an induced
P5 is exactly the class of (C4 , C5 , P5 )-free graphs. Since τc (C4 )/τ (C4 ) =
τc (P5 )/τ (P5 ) = 3/2, and τc (C5 )/τ (C5 ) = 4/3, any graph that contains C4 ,
C5 , or P5 as an induced subgraph does not satisfy the first property. Hence
it remains to show that every graph that does not satisfy the first property
contains either a C4 , a C5 , or a P5 as induced subgraph.
Consider a graph G = (V, E) with τc (G) > τ (G). If G is disconnected, we
consider one of its connected components such that every minimum vertex
cover of this connected component induces at least two connected components. So, we suppose that G is connected. Pick such a minimum vertex
cover C ⊂ V that induces the smallest number of connected components.
There must exist two subsets A, B ⊆ C inducing two disjoint connected
components, and a vertex v, such that G[A ∪ B ∪ {v}] is connected, by
Lemma 4.5.
Consider the breadth-first search (BFS) trees TA ⊆ E in G[{v} ∪ A],
and TB ⊆ E in G[{v} ∪ B], both rooted at v. If both trees have height
at least two, then there is an induced P5 . Hence at least one of the trees,
say TB , has height one, that is, N (v) ∩ B = B. Now we consider the set
C 0 := (C \ {w}) ∪ {v}, where w is an arbitrary vertex of B. Since the
number of connected components in G[C 0 ] is strictly less than the number of
connected components in G[C], and C 0 is not bigger than C, the new set C 0
cannot be a vertex cover. Therefore, there must exist a vertex x ∈
/ C, such
that wx ∈ E is not covered by C 0 . Note that xv ∈
/ E (otherwise it would not
be covered by C). If x is adjacent to a vertex t ∈ A that is itself adjacent
to v, then we have found a C4 . If x is adjacent to a vertex t ∈ A that is
not adjacent to v, then, using the shortest path from v to t in TA , we find a
cycle of length at least 5.
37
4.2 Structural results
Hence there remains the case where x is not adjacent to any vertex in A.
In that case, provided the height of TA is at least two, we can find a P5 . If
the height of TA is exactly one, then N (v) ∩ A = A, and we can do the same
reasoning as above, and show there is a vertex y ∈
/ C adjacent to a vertex
z ∈ A. Similarly, we can assume that y is not adjacent to any vertex in B.
Note that x 6= y because of non-adjacency between x and A and yz ∈ E
with z ∈ A. Hence, the path going from x to y through A, v, and B is an
induced P5 .
By this characterization, the problem of deciding whether τc (H)/τ (H) =
1 holds for every induced subgraph H of a given graph is solved in polynomial
time. In fact, this problem is equivalent to recognizing a P5 -free and chordal
graph. This problem is clearly solvable in polynomial time.
The above characterization tells us that the class of PoC-Perfect graphs
properly contains two well-known classes of graphs: split graphs and trivially
perfect graphs (see [26] for further reference on these classes). Moreover, it
gives rise to the following definition.
4.2.2
PoC-Near-Perfect Graphs
Let r ∈ [1, 2). A graph G is said to be PoC-Near-Perfect with threshold r
if every induced subgraph H of G satisfies τc (H) 6 r · τ (H). This defines a
hereditary class of graphs for every choice of r. Theorem 4.6 gives a forbidden
induced subgraph characterization of this class for r = 1. Our second result
gives such a characterization for r = 4/3.
Note that τc (C5 )/τ (C5 ) = 4/3 and τc (P5 )/τ (P5 ) = τc (C4 )/τ (C4 ) = 3/2.
Hence any graph class that does not forbid either C4 , C5 or P5 contains a
graph G such that τc (G)/τ (G) > 4/3. Therefore, the characterization of
Theorem 4.6 also holds for the class of graphs G such that every induced
subgraph H satisfies τc (H) 6 r · τ (H), for any r ∈ [1, 4/3). We now turn
our attention to r = 4/3, which is the next interesting threshold after r = 1.
Theorem 4.7. The following assertions are equivalent for every graph G:
(i) For every induced subgraph H of G it holds that τc (H) 6
4
3
· τ (H).
(ii) G is (P5 , C4 )-free.
To prove Theorem 4.7, we need to introduce the next notation and to
use the following lemma.
If G is not connected, there always exists a connected component of G
such that the price of connectivity of G is smaller than the price of connectivity of this connected component because a+b
c+d is in the non-oriented interval
a b
[ c , d ], for any a, b, c, d ∈ N0 . Therefore, we consider that G is connected. Let
C be a vertex cover of a graph G. Let C 0 be the vertex set of a connected
component of G[C]. We define PC (C 0 ) to be the set of vertices v ∈ V (G)
such that N [v] ∩ C ⊆ C 0 . It is clear that C 0 ⊆ PC (C 0 ).
38
Chapter 4. The price of connectivity for vertex cover
Lemma 4.8. Let S1 , S2 , . . . , Sk be the vertex sets of connected components
of a vertex cover C. There exists at least one PC (Si ) that is not a cutset of
G, i.e. G − PC (Si ) is connected.
Proof. We consider the graph H defined by
V (H) = {PC (Si )|i = 1, . . . , k}
and
E(H) = {PC (Si )PC (Sj )|N (PC (Si )) ∩ N (PC (Sj )) 6= ∅}.
Note that the sets PC (Si ), 1 6 i 6 k, are disjoint and induce a connected
subgraph of G each. Because C is a vertex cover, H is connected. Because
every connected graph contains a vertex without being a cutvertex, there
exists at least one PC (Si ) which is not a cutvertex of H. Therefore, PC (Si )
is not a cutset of G.
Proof of Theorem 4.7. Since the price of connectivity of P5 and C4 equals
3/2, any graph that contains C4 or P5 as an induced subgraph does not
satisfy the first property. Hence, it remains to show that every graph that
does not satisfy the first property contains either a C4 or a P5 as induced
subgraph.
Let G be a (P5 , C4 )-free graph. As explained before, we can suppose
that G is connected. We prove by induction on the number of connected
components of a minimum vertex cover, say k, that the price of connectivity
of G is bounded by 4/3. Let C be such a vertex cover of G. Let S1 , S2 , . . . , Sk
be the vertex set of the connected components of G[C].
If C is connected (k = 1), then τc /τ = 1.
If k = 2, i.e. S1 and S2 are connected components of G[C], we have a
vertex x adjacent to S1 and S2 , by Lemma 4.5. Hence, τc (G) 6 τ (G) + 1. If
G does not satisfy the first property of the theorem, τ (G) = 2, i.e. S1 = {s1 }
and S2 = {s2 }. If s1 and s2 have at least two common neighbors, there is
an induced C4 . Otherwise x is the unique common neighbor of s1 and s2 .
Because τc (G) 6= 2 = τ (G), each connected component of G[C] has at least
one private neighbor, i.e. there exist x1 , x2 ∈
/ C ∪ {x} with x1 s1 , x2 s2 ∈ E.
Note that x1 x2 ∈
/ E because C is a vertex cover and x1 , x2 ∈
/ C. Therefore
G[{x1 , s1 , x, s2 , x2 }] is an induced P5 .
If S1 , S2 and S3 are connected components of G[C], we can suppose,
without loss of generality, that there exist x1 ∈ N (S1 ) ∩ N (S2 ) and x2 ∈
N (S2 ) ∩ N (S3 ), by Lemma 4.5. The vertex x2 is adjacent to S1 (or x1 is
adjacent to S3 ), otherwise G[C ∪ {x1 , x2 }] contains an induced P5 . Thus,
C ∪ {x2 }, resp. C ∪ {x1 }, is a connected vertex cover. Hence,
τc (G)
|C| + 1
4
6
6 .
τ (G)
|C|
3
4.2 Structural results
39
Now k > 4 and we assume that τc 6 4/3τ holds for every connected
(P5 , C4 )-free graph with a minimum vertex cover of at most k − 3 connected components. Let S1 , S2 , S3 , . . . Sk be the vertex sets of the connected components of G[C]. By Lemma 4.8 at least one of these sets, say
PC (S3 ), is not a cutset of G. By applying twice Lemma 4.8, two more of
these sets, say PC (S2 ), resp. PC (S1 ), are not a cutset of G − PC (S3 ), resp.
G − (PC (S2 ) ∪ PC (S3 )). Let C 0 = C \ (S1 ∪ S2 ∪ S3 ) and note that C 0 is
a minimum vertex cover of G0 = G − (PC (S1 ) ∪ PC (S2 ) ∪ PC (S3 )). By the
induction hypothesis, there is a minimum connected vertex cover of G0 , say
Cc0 , with |Cc0 | 6 4/3|C 0 |.
We show that there exists a connected vertex cover Cc of G with |Cc | 6
|S1 | + |S2 | + |S3 | + |Cc0 | + 1, built from S1 , S2 , S3 and Cc0 . Indeed, we have
τc (G)
|S1 | + |S2 | + |S3 | + 1 + |Cc0 |
6
τ (G)
|S1 | + |S2 | + |S3 | + |C 0 |
|S1 | + |S2 | + |S3 | + 1 |Cc0 |
6 max
, 0
|S1 | + |S2 | + |S3 |
|C |
4
6 .
3
We refer to Cc0 as S4 for ease of writing. We observe that the set V (G) \ (S1 ∪
S2 ∪ S3 ∪ S4 ) is an independent set because its complement is a vertex cover
of G. We complete the proof with the following case distinction.
Case 1. There exists one component, say S1 , such that the other connected components are a distance 2 from S1 . Let xi be a vertex adjacent to
S1 and Si , for i = 2, 3, 4.
Case 1.1. It holds that x2 = x3 = x4 . We have immediately one vertex
to connect S1 , S2 , S3 and S4 .
Case 1.2. Two of the xi are equal, and the third one is distinct from
them. We can suppose without loss of generality that x3 = x4 . We suppose
by the previous case that x2 is not adjacent to S3 ∪ S4 and x3 is not adjacent
to S2 . The graph G[S2 ∪S1 ∪S3 ∪{x2 , x3 }] contains again a P5 not necessarily
induced. Thus, there must be an edge between x2 and S3 . But then, we have
an induced P5 in G[S2 ∪ S3 ∪ S4 ∪ {x2 , x3 }], a contradiction.
Case 1.3. The xi are mutually distinct and xi is not adjacent to Sj for
i 6= j (otherwise we are in the previous cases). Since G[S1 ∪S3 ∪S4 ∪{x3 , x4 }]
contains an induced P5 , we are in the next case.
Case 2. Up to a renaming of the Si , the distance between Si and Si+1 is
2, i = 1, 2, 3. Let xi be a vertex adjacent to Si and Si+1 , i = 1, 2, 3. Because
G is P5 -free, S1 must be adjacent to x2 or x1 must be adjacent to S3 . Hence,
we are in Case 1.
By Theorem 4.7, r = 3/2 is the next interesting threshold after r =
4/3. Our third result states that the list of forbidden induced subgraphs
for threshold r = 3/2 is (C6 , P7 , ∆1 , ∆2 ), where ∆1 is the flower made of
40
Chapter 4. The price of connectivity for vertex cover
two cycles of size 4 sharing a same vertex, and ∆2 is obtained from ∆1 by
removing one edge incident to the vertex of degree 4 (see Figure 4.2). A
flower is a connected graph G with only one cutvertex v such that G − v is
a linear forest and every path x1 · · · xk of G − v is connected to v by x1 and
xk .
Figure 4.2: Graphs ∆1 (on the left) and ∆2 (on the right).
Theorem 4.9. The following assertions are equivalent for every graph G:
(i) For every induced subgraph H of G it holds that τc (H) 6
3
2
· τ (H).
(ii) G is (P7 , C6 , ∆1 , ∆2 )-free.
To prove Theorem 4.9, we need to use the following lemma.
Lemma 4.10. Let G be a connected (C6 , P7 , ∆1 , ∆2 )-free graph, let C be a
vertex cover of G such that G[C] has exactly three connected components,
and let G contain an induced cycle of length 7 intersecting all connected
components of G[C]. Then there exists a connected vertex cover Cc such that
|Cc | 6 |C| + 1
|Cc | = 6
if |C| > 4,
if |C| = 4.
Proof. Let x1 x2 x3 x4 x5 x6 x7 be an induced cycle intersecting the three connected components of C, say S1 , S2 and S3 . Without loss of generality, we
can suppose x1 ∈ S1 , x3 ∈ S2 and x5 , x6 ∈ S3 (see Figure 4.3). We can
assume that no vertex is adjacent to S1 , S2 and S3 , otherwise the result is
obviously true.
x5
x6
x4
S3
x7
S2 x3
x1 S1
x2
Figure 4.3: G contains an induced C7 intersecting all components of a vertex
cover.
41
4.2 Structural results
If |N (x1 ) ∩ N (x3 )| > 2, |N (x1 ) ∩ N (x6 )| > 2, or |N (x5 ) ∩ N (x3 )| > 2,
then we have an induced subgraph ∆2 .
Otherwise N (x1 )∩N (x3 ) = {x2 } and N (x1 )∩N (x6 ) = {x7 } and N (x5 )∩
N (x3 ) = {x4 }. We distinguish several cases, depending on the cardinality of
S3 .
The first case is |S3 | = 2. If |S1 | = |S2 | = 1, i.e., |C| = 4, we have the
connected vertex cover S1 ∪ S2 ∪ S3 ∪ {x2 , x4 } of six vertices.
We can suppose |S2 | > 1. Then every vertex of S2 is adjacent to both x2
and x4 . Indeed, if an edge yz of S2 has one endpoint, say y, adjacent to both
x2 and x4 , then z must be adjacent to both x2 and x4 , otherwise G contains
an induced P7 . Let x be a vertex of S2 \ {x3 } which is not a cutvertex of
G[S2 ], i.e. Y = S1 ∪ S3 ∪ (S2 \ {x}) ∪ {x2 , x4 } induces a connected graph.
Such a vertex x always exists (for example, a leaf of a spanning tree of G[S2 ]
which is different from x3 ). If Y is not a vertex cover, there exists a vertex
t∈
/ Y adjacent to x. Note that t is distinct from x7 , because no vertex is
adjacent to S1 , S2 and S3 , and t is not adjacent to x1 or x5 , because G is
∆2 -free. Moreover t is not adjacent to x6 since G is C6 -free. Therefore we
have an induced P7 subgraph (see Figure 4.4).
x5
x6
x4
S2 x3
S3
x7
x
t
x1 S1
x2
Figure 4.4: G contains an induced C7 and P7 intersecting all components of
a vertex cover.
In the second case, |S3 | > 2. If there exists a vertex p1 in S3 \ {x5 , x6 }
which is adjacent to neither x4 nor x7 , we claim that there is an induced
P7 subgraph. Indeed, the distance in S3 ∪ {x4 , x7 } between this vertex p1
and x4 , resp. p1 and x7 , is at least 2. Without loss of generality, we suppose
that p1 is closer to x4 than x7 . Let p1 p2 · · · ps x4 be a shortest path between
p1 and x4 through S3 . By definition, for i = 1, . . . , s − 1, pi is adjacent
neither to x4 nor x7 . Note that ps is not adjacent to x7 because G is C6 -free.
Hence, G[{x7 , x1 , x2 , x3 , x4 , ps . . . , p2 , p1 }] contains an induced P7 . Let y ∈
S3 \{x5 , x6 } be such that y is not a cutvertex of G[S3 ]. Such a vertex y always
exists (for example, a leaf of a spanning tree rooted in x5 of G[S3 ] which is
different from x5 and x6 ). We can suppose that y is adjacent to x4 . Because
G is C6 -free, y is not adjacent to x7 . If Y = S1 ∪ S2 ∪ (S3 \ {y}) ∪ {x7 , x4 }
is not a vertex cover, there exists a vertex t ∈
/ Y adjacent to y. Note that
42
Chapter 4. The price of connectivity for vertex cover
t is distinct from x2 , because no vertex is adjacent to S1 , S2 and S3 (see
Figure 4.5). If t is adjacent to x1 (resp. x3 ), we have an induced C6 subgraph
(resp. ∆2 ).
x5
x6
y
x4
S2 x3
S3
x7
t
x1 S1
x2
Figure 4.5: Y = S1 ∪ S2 ∪ (S3 \ {y}) ∪ {x4 , x7 } is not a vertex cover of G.
Otherwise we have an induced P7 subgraph.
Proof of Theorem 4.9. If G contains one of the four forbidden induced subgraphs, say H, then τc (H)/τ (H) = 5/3. It remains to prove that the price
of connectivity of a (P7 , C6 , ∆1 , ∆2 )-free graph is bounded by 3/2. So let
G be a (P7 , C6 , ∆1 , ∆2 )-free graph. We can suppose that G is connected.
Otherwise we consider one of its connected components with the highest
price of connectivity. The proof is by induction on the number of connected
components of a minimum vertex cover. Let C be a minimum vertex cover
of G with the minimum number of connected components, say k.
If C is connected (k = 1), then τc /τ = 1.
If k = 2, by Lemma 4.5,
τc /τ 6
|C| + 1
1
1
3
61+
61+ = .
|C|
|C|
2
2
Now let k > 3. We may assume that τc 6 3/2τ holds for every (P7 , C6 , ∆1 ,
∆2 )-free graph with a minimum vertex cover of at most k − 2 connected
components. Let S1 , S2 , S3 , . . . , Sk be the vertex set of the connected components of G[C]. By Lemma 4.8, we may assume that the set PC (S2 ), is not
a cutset of G, and that the set PC (S1 ) is not a cutset of G0 = G − PC (S2 ).
Note that the set C 0 = C \ (S1 ∪ S2 ) is a minimum vertex cover of the
graph G00 = G − (PC (S1 ) ∪ PC (S2 )). By the induction hypothesis, there is a
minimum connected vertex cover of G00 , say Cc0 , with |Cc0 | 6 3/2|C 0 |.
We show that there exists a connected vertex cover Cc of G such that
|Cc | 6 |S1 | + |S2 | + |Cc0 | + 1, built from S1 , S2 and Cc0 . Indeed, we have
τc (G)
|S1 | + |S2 | + 1 + |Cc0 |
|S1 | + |S2 | + 1 |Cc0 |
3
6
6 max
, 0 6 .
0
τ (G)
|S1 | + |S2 | + |C |
|S1 | + |S2 |
|C |
2
We refer to the set Cc0 as S3 for the ease of writing. We can suppose that
there does not exist any single vertex to connect S1 , S2 , and S3 , otherwise
43
4.2 Structural results
S2
x1
S2
x2
S1
x1
S3
(a) Initial case
S1y
1
x2
y
z
z 3 S3
(b) Two private edges of a vertex cover
Figure 4.6: Three components of a vertex cover to connect by adding only
one vertex.
the result is obviously true. Without loss of generality, there is a vertex xi
adjacent only to Si and Si+1 , i = 1, 2, such that x1 6= x2 (see Figure 4.6a).
Note that x1 and x2 are not adjacent because C is a vertex cover. Let
y1 ∈ S1 and z3 ∈ S3 be two vertices such that y1 and z3 are not cutvertices
in G[S1 ∪ S2 ∪ S3 ∪ {x1 , x2 }]. If S1 ∪ S2 ∪ (S3 \ {z3 }) ∪ {x1 , x2 } or (S1 \
{y1 }) ∪ S2 ∪ S3 ∪ {x1 , x2 } is a vertex cover, then τc (G)/τ (G) 6 3/2. Thus,
there exist two edges, say y1 y and z3 z, with y, z ∈
/ S1 ∪ S2 ∪ S3 ∪ {x1 , x2 }
(see Figure 4.6b). Note that y can be equal to z.
Now, we discuss the adjacency of y with S3 and S2 .
Case 1. The vertex y is adjacent to S3 . Thus, y is not adjacent to S2 . If
the shortest induced cycle via S1 ∪ S2 ∪ S3 ∪ {x1 , x2 , y} is of length 6 or more
than 8, we have an induced C6 or a P7 . Thus the shortest induced cycle via
the three connected components has 7 vertices. By Lemma 4.10, it is clear
that τc (G)/τ (G) 6 3/2 if |S1 | + |S2 | + |S3 | > 4. Otherwise S1 , S2 and S3 are
three connected components of the initial vertex cover C. Thus, by Lemma
4.10, τc (G)/τ (G) 6 6/4 = 3/2.
The cases that z is adjacent to S1 or y = z are dealt with similarly.
Case 2. The vertex y is adjacent to S2 and z is not adjacent to S2 . Since
G[S1 ∪ S2 ∪ S3 ∪ {x1 , x2 , y, z}] does not contain an induced P7 , there exists
t ∈ N (x1 ) ∩ N (x2 ) ∩ S2 and t is adjacent to y. Hence, we have an induced
∆2 .
Case 3. Both y and z are adjacent to S2 . Thus y (resp. z) is not
adjacent to S3 (resp. S1 ). Let P be a shortest path from z to y that goes
through S3 , {x2 }, S2 , {x1 }, and S1 . If P has 7 vertices, then we have an
induced P7 , ∆1 or ∆2 subgraph, depending on the adjacency of y and z with
S2 . If P contains at least nine vertices, we have an induced P7 subgraph in
G[S1 ∪ S2 ∪ S3 ∪ {x1 , x2 }]. Otherwise P has exactly 8 vertices. There are
two possibilities.
Case 3.1. S1 (or S3 ) contains an edge of P (see Figure 4.7a). Thus we
have an induced P7 or ∆2 in G[{z, x1 , x2 } ∪ S1 ∪ S2 ∪ S3 }], depending on the
adjacency between S2 and z.
Case 3.2. S2 contains an edge of P (see Figure 4.7b), say vu. Then, if
z is not adjacent to v, G contains a P7 or a ∆2 , depending on the adjacency
44
Chapter 4. The price of connectivity for vertex cover
S2
v
x1
S1y
x2
y
z
1
S2
u
x1
z3S3
(a) Case 3.1.
S1
x2
y
z
z3
y1
S3
(b) Case 3.2.
Figure 4.7: Three components of a vertex cover to connect by adding only
one vertex.
between z and u. Thus z is adjacent to v. Hence, we have an induced P7
or ∆2 subgraph in G[S1 ∪ (S2 \ {u}) ∪ S3 ∪ {x1 , x2 , y, z}], depending on the
adjacency between y and v.
Case 4. The vertex y is adjacent to neither S2 nor S3 . We can suppose
that z is adjacent to neither S1 nor S2 (thus y 6= z). Thus, G contains an
induced P7 .
Since a chordal and P7 -free graph is (C6 , P7 , ∆1 , ∆2 )-free, we deduce the
following corollary from Theorem 4.9.
Corollary 4.11. If G is a chordal, P7 -free graph, then for every induced
subgraph H of G, it holds that τc (H) 6 3/2 · τ (H).
4.2.3
PoC-Critical Graphs
We now turn our attention to critical graphs, that is, graphs G for which
the price of connectivity of any proper induced subgraph H of G is strictly
smaller than the price of connectivity of G. These are exactly the graphs
that can appear in a forbidden induced subgraph characterization of the PoCnear-perfect graphs for some threshold r ∈ [1, 2). A perhaps more tractable
class of graphs is the class of strongly critical graphs, defined as the graphs
G for which every proper (not necessarily induced) subgraph H of G has a
price of connectivity that is strictly smaller than the price of connectivity of
G. It is clear that every strongly critical graph is critical, but the converse
is not true. For instance, C5 is critical, but not strongly critical. Notice that
every (strongly) critical graph is connected.
PoC-Critical Chordal Graphs: Let T be a tree. We call T special if
it is obtained from another tree by subdividing each edge exactly once and
then attaching a pendent vertex to every leaf of the resulting graph (see
Figure 4.8 for an example).
Our following result characterizes the class of (strongly) critical chordal
graphs.
4.2 Structural results
45
Figure 4.8: A special tree constructed from another tree (vertices indicated
by filled circles) by subdividing each edge exactly once (subdivision vertices
are indicated by hollow circles) and by attaching a pendent vertex (indicated
by squares) to every leaf of the resulting graph.
Theorem 4.12. For a chordal graph G, the following assertions are equivalent:
(i) G is a special tree.
(ii) G is strongly critical.
(iii) G is critical.
Lemma 4.13. Let G be a critical graph. For every minimum vertex cover
C of G, there does not exist a bridge of G with endpoints in C.
Proof. Suppose there exists a bridge xy with x, y ∈ C. The removal of the
edge xy results in two connected subgraphs of G, which we denote by G1
resp. G2 . We can assume that x ∈ V (G1 ) and y ∈ V (G2 ). Let G01 be the
graph obtained from G1 by attaching a pendent vertex to x. Similarly let
G02 be the graph obtained from G2 by attaching a pendent vertex to y.
We observe that C ∩ V (G1 ) is a vertex cover of G01 and C ∩ V (G2 ) is a
vertex cover of G02 . Thus
τ (G) > τ (G01 ) + τ (G02 ).
(4.2)
On the other hand, let Cc,1 be a minimum connected vertex cover of G01
and Cc,2 be a minimum connected vertex cover of G02 . We can assume that
Cc,1 ⊆ V (G1 ) and Cc,2 ⊆ V (G2 ). It is clear that x ∈ Cc,1 and y ∈ Cc,2 . Thus
Cc,1 ∪ Cc,2 is a connected vertex cover of G. Since Cc,1 ∩ Cc,2 = ∅,
τc (G) 6 τc (G01 ) + τc (G02 ).
(4.3)
But (4.2) and (4.3) say that
τc (G)/τ (G) 6 max{τc (G01 )/τ (G01 ), τc (G02 )/τ (G02 )}.
(4.4)
Since both G01 and G02 are isomorphic to induced subgraphs of G, (4.4) is a
contradiction to the fact that G is critical.
Proof of Theorem 4.12. It is obvious that (ii) implies (iii). First, we prove
that (iii) implies (i), that is, every critical chordal graph is a special tree.
For this, let G be a critical chordal graph. Hence, G is connected.
46
Chapter 4. The price of connectivity for vertex cover
If the chordal graph G is not a tree, then G contains a triangle as a
subgraph and every minimum vertex cover of G contains at least two vertices
of this triangle. Let v be a vertex that is both in the triangle and in a
minimum vertex cover. Then we have τ (G) = τ (G − v) + 1 and also τc (G) 6
τc (G − v) + 1, implying that G is not critical. Therefore, G is a tree.
Let C be a minimum vertex cover of G.
First we show that C is an independent set. Suppose there are x, y ∈ C
such that xy ∈ E. Since G is a tree, xy is a bridge, a contradiction with
Lemma 4.13.
Now we show that every member of V \ C has degree at most two. For
this, let x ∈ V \ C. Suppose that |N (x)| > 3. Let X1 , X2 , . . . , Xk be the
vertex sets of the connected components of G − x. By assumption, k > 3.
Let
k
[
H1 = G −
Xi
i=3
and
H2 = G − (X1 ∪ X2 ).
We observe that
τ (G) > τ (H1 ) + τ (H2 ).
(4.5)
Since x is a cutvertex of H1 , x is contained in every connected vertex
cover of H1 . Therefore
τc (G) 6 τc (H1 ) + τc (H2 ).
(4.6)
By the same argumentation from Lemma 4.13, (4.5) and (4.6) yield a
contradiction to the fact that G is critical. This proves that every vertex
of V \ C has at most two neighbors. By the discussion above, C is an
independent set and G is a tree. Moreover, the degree of every vertex in C
is at least two. Otherwise let v be a vertex of C with degree 1 and let u be
the neighbor of v. Because C is independent, u ∈
/ C. Because C is a vertex
cover, every neighbor of u is in C. Thus, Y = (C \ {v}) ∪ {u} is a minimum
vertex cover but Y is not independent, a contradiction. We prove that G is
a special tree. In fact, the initial tree H is defined as followed: V (H) = C
and E(H) = {uv|there exists a path Puv in (V (G) \ C) ∪ {v, u} from u to v}.
Because C is a vertex cover of G, if uv is an edge in H, then the length of
the path Puv in G is exactly 2. Moreover, two degree-1 vertices, say x and y,
cannot have the same neighbor since τc (G−x) = τc (G), τ (G−x) = τ (G) and
G is critical. The neighbor y of any degree-1 vertex x must have its degree
equal to 2. Otherwise, clearly τ (G − x) 6 τ (G). The vertex y is a cutvertex
of the tree G − x, hence τc (G − x) = τc (G). We obtain a contradiction since
G is critical. All in all, G is a special tree.
Now, we show that (i) implies (ii), that is, every special tree is strongly
critical. Let G be a special tree. It is easy to see that τc (G)/τ (G) = 2 −
4.2 Structural results
47
1/τ (G). If G is not strongly critical, then there exists a proper subgraph H
of G such that τc (H)/τ (H) > τc (G)/τ (G). We can suppose that such an H
is minimal for inclusion. Thus H is critical. By the previous argumentation,
H is a special tree. Therefore, τc (H)/τ (H) = 2 − 1/τ (H), but 2 − 1/τ (G) >
2 − 1/τ (H 0 ) for every proper special subtree H 0 of G, a contradiction. This
completes the proof.
PoC-Strongly-Critical Graphs: Our following result yields structural
constraints on the class of strongly critical graphs.
Theorem 4.14. Let G be a strongly critical graph.
(i) Every minimum vertex cover of G is independent. In particular, G is
bipartite.
(ii) If G has a cutvertex, then G is a special tree.
Theorem 4.14 follows from Lemma 4.15 and Lemma 4.16 presented below.
Lemma 4.15. Let G be a strongly critical graph. Then every minimum
vertex cover of G is an independent set. In particular, G is bipartite.
Proof. Let G be a strongly critical graph and let C be a minimum vertex
cover of G. We know that G is connected. Suppose that C is not an independent set. Thus, there are two adjacent vertices in C, say x and y.
By Lemma 4.13, xy cannot be a bridge of G. So G − xy is connected.
Let Cc be a minimum connected vertex cover of G − xy. Suppose that
{x, y} ∩ Cc 6= ∅. Then τc (G − xy) = τc (G), a contradiction with G strongly
critical. Thus {x, y} ∩ Cc = ∅. Hence Cc ∪ {x} is a connected vertex cover of
G with y ∈
/ Cc ∪ {x}. Moreover, Cc ∪ {x} is minimum since every minimum
connected vertex cover of G − xy contains neither x nor y.
We go back to the vertex cover C, let A = NG (y)∩C and B = NG (y)\C.
As x ∈ A, A 6= ∅. Since C is a minimum vertex cover and y ∈ C, B 6= ∅. Let
G0 be the graph obtained from G by the removal of all edges joining y to B.
Since Cc ∪ {x} is a connected vertex cover of G and y ∈
/ Cc , then NG (y) =
A ∪ B ⊆ Cc ∪ {x} and G0 is connected. As C \ {y} is a vertex cover of G0 ,
τ (G0 ) < τ (G). We prove that τc (G0 ) 6 τc (G) − 2. If τc (G0 ) > τc (G), we have
a contradiction since G is strongly critical. Otherwise τc (G0 ) = τc (G) − 1.
Then
τc (G0 )
τc (G0 ) + 1
τc (G)
τc (G)
>
=
>
,
0
0
0
τ (G )
τ (G ) + 1
τ (G ) + 1
τ (G)
a contradiction because G is strongly critical. Let Cc0 be a minimum connected vertex cover of G0 . To cover the edge xy, either x or y is in Cc0 . If
x∈
/ Cc0 , then y and at least one vertex from NG0 (y) = A are in Cc0 by connectivity. In each case, A ∩ Cc0 6= ∅. Therefore Cc0 ∪ {y} is a connected vertex
cover of G, a contradiction to the fact that |Cc0 ∪{y}| 6 τc (G0 )+1 6 τc (G)−1.
This completes the proof.
48
Chapter 4. The price of connectivity for vertex cover
Lemma 4.16. Let G be a strongly critical graph. If G has a cutvertex, it is
a special tree.
Proof. Let G = (V, E) be a strongly critical graph with a cutvertex. We
know that G is connected. Suppose that G is not a tree. Thus G has a nontrivial block. We can pick a cutvertex x and an edge e incident to x in this
block. The graph G − e is connected, by the choice of e. Every connected
vertex cover of G − e contains x, as x is a cutvertex of G − e. Hence,
every connected vertex cover of G − e covers e. Thus τc (G − e) > τc (G), a
contradiction with G strongly critical. Hence, G is a tree. In particular, G
is chordal.
The conclusion then follows from Theorem 4.12.
4.2.4
Aside: values of the price of connectivity
In this section, we answer a natural question: for which value r ∈ [1, 2),
does there exist a graph whose price of connectivity is exactly r? Of course,
we are interested only in rational values since the price of connectivity is
defined by a ratio between two integers. The following theorem shows that
for each rational value r, there exists a (P6 , C6 )-free graph G such that
τc (G)/τ (G) = r.
Theorem 4.17. Let r be a rational number in [1, 2). There exists a (P6 , C6 )free graph G such that τc (G)/τ (G) = r.
Proof. Assume that r = pq , with p and q two integers. Let s = p−q. Consider
Fp,q the graph built from the flower, made of s cycles on 4 vertices sharing a
same vertex v, with q − s − 1 paths of length 2 attached to v, as illustrated
by Figure 4.9. Clearly, Fp,q is (P6 , C6 )-free.
Figure 4.9: The set of black vertices indicates a minimum vertex cover (respectively a minimum connected vertex cover) of F8,5 .
We will prove that τ (Fp,q ) = q and τc (Fp,q ) = p. Note that s < q since
r < 2. Hence, q − 1 − s > 0. A minimum vertex cover consists of the center
and all vertices at distance 2 from the center, as illustrated by Figure 4.9.
This implies that τ (Fp,q ) = s + (q − 1 − s) + 1 = q. Furthermore, a minimum
connected vertex cover consists of all vertices of degree > 1 except one vertex
for each petal, which means that τc (Fp,q ) = 2s + 1 + q − 1 − s = p.
Chapter 5
The price of connectivity for
domination
In parallel to the study of the price of connectivity for the vertex cover
problem, let us examine the interdependence of the domination number γ
and the connected domination number γc both from a complexity-theoretic
point of view and in some hereditary classes of graphs. This chapter is
partly based on a paper published in Discrete Applied Mathematics [36].
Many proofs in this chapter are similar to those of the previous chapter.
Duchet and Meyniel [55] observed that for every graph G it holds that
γc (G) 6 3γ(G) − 2. As an immediate consequence, every graph G satisfies
γc (G)/γ(G) < 3,
(5.1)
that is, the price of connectivity for the dominating set problem of any graph
G is strictly bounded by 3.
Our starting point is the following result by Zverovich [126] on the class of
(P5 , C5 )-free graphs, wherein the domination is also N P -complete, as proved
by Bertossi [19], and by Corneil and Perl [48].
Theorem 5.1 (Zverovich [126]). The following assertions are equivalent for
every graph G.
(i) For every induced subgraph H of G it holds that γc (H) = γ(H).
(ii) G is (P5 , C5 )-free.
Firstly, we study the computational complexity of the problem of deciding
whether the price of connectivity of a graph given as input is bounded by a
certain constant r.
Secondly, we prove bounds on the price of connectivity in the class of
(Pk , Ck )-free graphs for k > 6. Actually, we study classes of graphs whose
price of connectivity for any induced subgraphs is bounded by a constant
49
50
Chapter 5. The price of connectivity for domination
r ∈ (1, 3). For a small value of r, those classes will be defined by a finite list
of forbidden induced subgraphs. Hence, the forbidden subgraph characterizations directly yield polynomial-time algorithms for recognizing graphs in
those classes.
Finally, we bound the price of connectivity depending on the radius or
the diameter of the graph. Moreover, for any rational number r ∈ [1, 3), we
find a (P9 , C9 )-free graph with price of connectivity exactly r.
5.1
Computational Complexity
The class ΘP2 , sometimes denoted P NP[log] , is defined as the class of decision
problems solvable in polynomial time by a deterministic Turing machine that
allows using O(log n) many queries to an N P -oracle, where n is the size of
the input.
Theorem 5.2. Let 1 < r < 3 be a fixed rational number. Given a connected
graph G, the problem of deciding whether γc (G)/γ(G) 6 r is Θp2 -complete.
It is easy to see that the above decision problem belongs to Θp2 , since
both γ(G) and γc (G) can be computed using logarithmically many queries
to an N P -oracle by binary search. Thus, Theorem 5.2 is a negative result:
loosely speaking, it tells us that deciding whether the ratio of γc (G) and
γ(G) is bounded by a certain constant is as hard as computing both γc (G)
and γ(G) explicitly, and this remains true even if the constant is not part of
the input.
Our reduction is from the decision problem whether for two given graphs
G and H it holds that τ (G) > τ (H), which is known to be Θp2 -complete due
to Spakowski and Vogel [116]. To simplify the construction, we may, without
loss of generality, assume that G and H are both connected: otherwise we
may add an isolated vertex to G and then a dominating vertex, and proceed
similarly with H. Denoting the resulting graphs by G0 and H 0 , we see that
τ (G0 ) = τ (G) + 1 and τ (H 0 ) = τ (H) + 1.
Before proving Theorem 5.2, we need the following two lemmas.
Lemma 5.3. Given a connected graph G with n > 2 vertices, one can construct in linear time a graph G0 such that γ(G0 ) = n + τ (G) and γc (G0 ) =
3n − 1.
Proof. To each vertex v ∈ V (G), associate three vertices v, v 0 , v 00 in V (G0 ).
Moreover, for each edge e ∈ E(G), define a vertex e of V (G0 ). We may treat
V (G) and E(G) as subsets of V (G0 ). Let
E(G0 ) :=
[
{ue, ve} ∪
e=uv∈E(G)
[
v∈V (G)
{vv 0 , v 0 v 00 }.
5.1 Computational Complexity
51
Let D be a minimum dominating set of G0 . Since every vertex of the
form v 00 is of degree 1, we can assume that v 0 ∈ D for every v ∈ V (G).
Assume that e ∈ D for some e = uv ∈ E(G). Since u0 , v 0 ∈ D, the set
D \ {e} dominates all vertices of G0 except e. Thus (D \ {e}) ∪ {u} is a
minimum dominating set of G0 . So we may assume D ∩ E(G) = ∅. Hence,
D ∩ V (G) is a vertex cover of G, and so γ(G0 ) = |D| > n + τ (G).
Similarly, if T is a vertex cover of G, the set {v 0 | v ∈ V (G)} ∪ T is a
dominating set of G. Hence, γ(G0 ) 6 n + τ (G). This gives γ(G0 ) = n + τ (G).
To see that γc (G0 ) = 3n − 1, let C be a minimum connected dominating
set of G0 . Recall that n > 2 and G contains at least one edge. Thus, the
vertices v and v 0 , for every v ∈ V (G), are cutvertices of G0 and therefore
contained in C. Moreover, C contains no vertex of the form v 00 .
Observe that the set C ∩E(G) defines a minimum spanning tree of G, and
so |C ∩E(G)| = n−1. Summarizing, γc (G0 ) = |C| = 2n+n−1 = 3n−1.
Lemma 5.4. Given a graph G with n vertices and m edges, one can construct
in linear time a graph G0 such that γ(G0 ) = n + m + 1 and γc (G0 ) = n + m +
1 + τ (G).
Proof. We start from the construction given in the proof of Lemma 5.3. We
add two new vertices w, w0 and, for every e ∈ E(G), a vertex e0 . Then we
put edges joining w to every vertex of the form v 0 , where v ∈ V (G), and to
w0 . We also put an edge joining e and e0 , for every e ∈ E(G). The resulting
graph is called G0 .
For each edge e ∈ E(G), the corresponding vertex e ∈ V (G0 ) is adjacent
to the degree-1 vertex e0 . Thus, we can consider without loss of generality
only the minimum dominating sets of G0 that contain all vertices from E(G).
The same remark holds for every vertex v 0 , where v ∈ V (G), and for w. Now
the set D ⊆ V (G0 ) of those vertices is a dominating set of G0 , hence we have
γ(G0 ) = n + m + 1.
γc (G0 ) remains to be computed. The previous dominating set D is not
connected, as G0 [D] has exactly m + 1 connected components: one for each
edge of G, and one induced by w and the vertices of the form v 0 . To make D
connected, we need to add the fewest possible additional vertices v ∈ V (G).
Every such vertex v will link the component containing v to every vertex
e ∈ E(G) of G0 such that v ∈ e. Hence the minimum number of additional
vertices to add to C is exactly the size τ (G) of a minimum vertex cover of
G. Hence γc (G0 ) = n + m + 1 + τ (G).
Proof of Theorem 5.2. Let r = r1 /r2 be a fixed rational number with 1 <
r < 3. We have already argued why the decision problem is in Θp2 , so we
proceed to proving the Θp2 -hardness. Let G and H be two graphs. We reduce from the Θp2 -complete decision problem of deciding whether τ (G) >
τ (H) [116]. We may, without loss of generality, assume that G and H are
52
Chapter 5. The price of connectivity for domination
both connected, as described previously. The reduction consists of the following five steps.
Step 1. We choose an arbitrary vertex v ∈ V (G). Starting with r2 disjoint
copies of G, we connect all r2 copies of v to a new vertex w. We then
attach a pendent vertex w0 to w. We denote by Gr2 the resulting graph. Let
nG = |V (G)|. Clearly, τ (Gr2 ) = r2 τ (G) + 1 and |V (Gr2 )| = r2 nG + 2.
Similarly we construct Hr1 from H. Let nH = |V (H)| and mH = |E(H)|.
Clearly, τ (Hr1 ) = r1 τ (H) + 1, |V (Hr1 )| = r1 nH + 2, and |E(Hr1 )| = r1 mH +
r1 + 1.
Step 2. We apply Lemma 5.3 to Gr2 to get G0r2 . The following results are
obtained:
γ(G0r2 ) = |V (Gr2 )| + τ (Gr2 )
= r2 τ (G) + r2 nG + 3,
γc (G0r2 )
= 3|V (Gr2 )| − 1
= 3r2 nG + 5.
Then, we apply Lemma 5.4 to Hr1 to get Hr0 1 , and obtain
γ(Hr0 1 ) = |V (Hr1 )| + |E(Hr1 )| + 1
= r1 (nH + mH + 1) + 4,
γc (Hr0 1 )
= τ (Hr1 ) + |V (Hr1 )| + |E(Hr1 )| + 1
= r1 τ (H) + r1 (nH + mH + 1) + 5.
Step 3. We construct a new graph U by taking the disjoint union of two
copies of G0r2 and two copies of Hr0 1 , picking a vertex from each of these four
graphs that is adjacent to a degree-1 vertex, and then adding any possible
edge between these four vertices. Observe that there exists a minimum
connected dominating set in each of the four copies containing the picked
vertex.
By the construction of U ,
γc (U ) = 2γc (G0r2 ) + 2γc (Hr0 1 )
= 2r1 τ (H) + 2(r1 (nH + mH + 1) + 3r2 nG + 10),
γ(U ) = 2γ(G0r2 ) + 2γ(Hr0 1 )
= 2r2 τ (G) + 2(r1 (nH + mH + 1) + r2 nG + 7).
Step 4. Let
ϕ1 = r1 (nH + mH + 1) + 3r2 nG + 10,
ϕ2 = r1 (nH + mH + 1) + r2 nG + 7.
53
5.2 Structural results
Let p = max{|ϕ1 − 3ϕ2 |, |ϕ2 − ϕ1 |}, and
a = p(3r2 − r1 ) + (ϕ1 − 3ϕ2 ),
b = p(r1 − r2 ) + (ϕ2 − ϕ1 ).
By definition of p, a > |ϕ1 − 3ϕ2 |(3r2 − r1 ) + (ϕ1 − 3ϕ2 ) > |ϕ1 − 3ϕ2 |(3r2 −
r1 − 1) and b > |ϕ2 − ϕ1 |(r1 − r2 ) + (ϕ2 − ϕ1 ) > |ϕ2 − ϕ1 |(r1 − r2 − 1). Since
r1 > r2 and 3r2 > r1 , then a and b are non-negative integers. Moreover,
a, b ∈ O(ϕ1 + ϕ2 ). An easy calculation shows that
a + 3b + 2ϕ1 = 2pr1 and a + b + 2ϕ2 = 2pr2 .
(5.2)
We now construct a graph U 0 from U as follows. Let v be a degree-1
vertex in U (such a vertex is always present). Let P a be the graph obtained from the chordless path with vertex set {u1 , u2 , . . . , ua } by attaching
a pendent vertex to every member of {u1 , u2 , . . . , ua }. Let P b be the graph
obtained from the chordless path with vertex set {v1 , v2 , . . . , v3b } by attaching a pendent vertex to every member of {v3 , v6 , . . . , v3b }. Let U 0 be the
graph obtained from the disjoint union of U , P a and P b by putting an edge
from v to u1 and to v1 . Since a, b ∈ O(ϕ1 + ϕ2 ), the graph U 0 can be
constructed from G and H in linear time.
By the construction of U 0 , we obtain
γc (U 0 ) = γc (U ) + a + 3b
= 2r1 τ (H) + a + 3b + 2ϕ1
(5.2)
= 2r1 τ (H) + 2pr1 ,
and
γ(U 0 ) = γ(U ) + a + b
= 2r2 τ (G) + a + b + 2ϕ2
(5.2)
= 2r2 τ (G) + 2pr2 .
Recalling r = r1 /r2 , we have
γc (U 0 )
2r1 τ (H) + a + 3b + 2ϕ1
2r1 τ (H) + 2pr1
τ (H) + p
=
=
=r
.
0
γ(U )
2r2 τ (G) + a + b + 2ϕ2
2r2 τ (G) + 2pr2
τ (G) + p
Thus, γc (U 0 )/γ(U 0 ) 6 r if and only if τ (H) 6 τ (G). This completes the
proof.
5.2
Structural results
Before establishing further results, we prove the following lemma. Note that
this lemma is concerned with minimal connected dominating sets.
54
Chapter 5. The price of connectivity for domination
Lemma 5.5. Let G be a connected graph that is (Pk , Ck )-free for some k > 4
and let X be a minimal connected dominating set of G. Then G[X] is Pk−2 free.
Proof. Suppose that there is an induced path v1 v2 · · · vk−2 on k − 2 vertices
in G[X]. As X is minimal, X \ {v1 } is not a connected dominating set.
Hence, X \ {v1 } is not a dominating set or G[X \ {v1 }] is disconnected. In
the first case, there is a vertex v10 ∈ V \ X whose only neighbor in X is v1 . In
the second case, the vertices v2 , . . . , vk−2 are contained in a single connected
component of G[X \{v1 }]. Thus, there is a neighbor of v1 in X, say v10 , that is
not adjacent to any member of {v2 , . . . , vk−2 }. In both cases, there is a vertex
v10 ∈
/ {v1 , v2 , . . . , vk−2 } whose only neighbor among {v1 , v2 , . . . , vk−2 } is v1 .
0
Similarly, there is a vertex vk−2
∈
/ {v1 , v2 , . . . , vk−2 } whose only neighbor
0
among {v1 , v2 , . . . , vk−2 } is vk−2 . But then G[{v10 , v1 , v2 , . . . , vk−2 , vk−2
}] is
0
0
isomorphic to Pk or to Ck , depending on the adjacency of v1 and vk−2 , and
this is a contradiction.
5.2.1
PoC-Near-Perfect Graphs
As Theorem 5.2 shows, the class of connected graphs where γc (G)/γ(G) 6 r
holds (for any fixed rational r ∈ (1, 3)) is Θp2 -complete to recognize. However,
if we restrict our attention to hereditary graph classes, we are able to derive
the following results. Note that our characterizations yield polynomial-time
recognition algorithms, since the list of forbidden induced subgraphs is finite
in each case.
We know from Zverovich [126] that the hereditary class of graphs G for
which γc (G) = γ(G), referred to as PoC-perfect graphs, is exactly the set of
(P5 , C5 )-free graphs.
Let r ∈ [1, 3). A graph G is said to be PoC-near-perfect with threshold
r if every induced subgraph H of G satisfies γc (H) 6 r · γ(H). This defines a hereditary class of graphs for every choice of r. Theorem 5.1 from
Zverovich [126] gives a forbidden induced subgraph characterization of this
class for r = 1. Our second result gives indirectly such a characterization for
r = 3/2.
Note that γc (C5 )/γ(C5 ) = γc (P5 )/γ(P5 ) = 3/2. Hence any graph class
that does not forbid either C5 or P5 contains a graph G such that γc (G)/γ(G)
= 3/2. Therefore, the characterization of Theorem 5.1 also holds for the class
of graphs G such that every induced subgraph H satisfies γc (H) 6 r · γ(H),
for any r ∈ [1, 3/2). We now turn our attention to r = 3/2, which is the
next interesting threshold after r = 1.
Our first result establishes basically the upper bound γc (G) 6 γ(G) + 1
for every connected (P6 , C6 )-free graph G. We turn this result into a characterization for the class of (P6 , C6 )-free graphs by holding the upper bound
inequality for every connected induced subgraph. Moreover, this character-
55
5.2 Structural results
ization gives the characterization of PoC-near-perfect graphs with threshold 3/2, as a corollary. However, γc (G) 6 γ(G) + 1 for every connected
(P6 , C6 )-free graph G is a stronger result than γc (G)/γ(G) 6 3/2 for every
(P6 , C6 )-free graph G.
Theorem 5.6. For every graph G, the following assertions are equivalent:
(a) For every connected induced subgraph H of G it holds that γc (H) 6
γ(H) + 1.
(b) G is (P6 , C6 )-free.
Proof. It is straightforward to check that γc (P6 ) = γc (C6 ) = 4 and γ(P6 ) =
γ(C6 ) = 2. Thus, (a) implies (b).
Now let G = (V, E) be a connected (P6 , C6 )-free graph and let H be a
connected induced subgraph of G. Observe that H is (P6 , C6 )-free. To see
that (b) implies (a), it suffices to prove that γc (H) 6 γ(H) + 1. For this,
let D be a minimum dominating set of H. Let D1 , D2 , . . . , Dk be the vertex
sets of the connected components of H[D]. Let C ⊆ V be an inclusion-wise
minimal set such that H[D ∪ C] is connected, and let X ⊆ D ∪ C be a
minimal connected dominating set of H. By Lemma 5.5, H[X] is P4 -free.
Let I ⊆ {1, 2, . . . , k} be such that i ∈ I if and only if Di ∩ X = ∅. For
every i ∈ I, pick xi ∈ X such that xi has a neighbor in Di (this is always
possible, since X is a dominating set). Note that every xi belongs to C,
and that the xi ’s do not have to be distinct. The situation is illustrated by
Figure 5.1.
D1
x1
D2
D3
C
x3
x5
D4
D5
D6
Figure 5.1: D ∪ C is a minimal connected dominating set containing D but
X, representing by the gray areas, is a minimal connected dominating set
included in D ∪ C. Here, x5 = x6 .
Let S =
S
i∈I
/ (Di
∩ X) ∪ {xj | j ∈ I}. To estimate the cardinality of X,
56
Chapter 5. The price of connectivity for domination
we will prove that S is nearly another description of the dominating set X,
depending on the connectivity of H[S].
Assume first that H[S] is connected. Then H[D ∪ {xi | i ∈ I}] is connected, and so C = {xi | i ∈ I}, since C was chosen minimal such that
H[D ∪ C] is connected. Thus, X = S, which gives
X
γc (H) 6 |X| = |S| 6
|Di ∩ X| + |I| 6 |D| = γ(H).
i∈I
/
Now assume that H[S] is not connected. Seinsche [113] proved that every P4 -free graph with at least two vertices is either disconnected, or its
complement is disconnected. In particular, this applies to the connected
graph H[X], i.e. the complement H[X] of H[X] is disconnected. Since the
complement of H[S] is connected, the graph H[X] has a connected component containing S and at least one other connected component Y . In H,
every vertex in V (Y ) is adjacent to every vertex in S. Let y ∈ V (Y ). Then
H[D ∪ {xi | i ∈ I} ∪ {y}] is connected, and so C = {xi | i ∈ I} ∪ {y}, since C
was chosen minimal such that H[D ∪ C] is connected. Thus, X = S ∪ {y},
which gives
X
γc (H) 6 |X| = |S| + 1 6
|Di ∩ X| + |I| + 1 6 |D| + 1 = γ(H) + 1.
i∈I
/
This completes the proof.
To see that the bound given by Theorem 5.6 is the best possible one,
consider the infinite family {Fk | k > 2} of (P6 , C6 )-free graphs where Fk
is the graph obtained from K1,k by subdividing each edge exactly once, as
illustrated by Figure 5.2. Clearly γc (Fk ) = k + 1 = γ(Fk ) + 1.
Figure 5.2: The set of black vertices indicates a minimum dominating set
(resp. a minimum connected dominating set) of F4 .
From Theorem 5.6 and because γc (P6 )/γ(P6 ) = γc (C6 )/γ(C6 ) = 2, we
deduce the following corollary which gives exactly the characterization of
PoC-near-perfect graphs with threshold 3/2.
Corollary 5.7. For every graph G, the following assertions are equivalent:
5.2 Structural results
57
(a) For every induced subgraph H of G it holds that γc (H)/γ(H) 6 23 .
(b) G is (P6 , C6 )-free.
Notice that the assumption in Corollary 5.7 does not required that the
graph G is connected. Indeed, if G is not connected, there always exists a
connected component of G such that the price of connectivity of G is smaller
than the price of connectivity of this connected component because a+b
c+d is
a b
in the non-oriented interval [ c , d ], for any a, b, c, d ∈ N0 .
By Corollary 5.7, r = 2 is the next interesting threshold after r = 3/2.
Our second result states a sufficient condition for threshold r = 2: being a
(C8 , P8 )-free graph.
Theorem 5.8. For every (P8 , C8 )-free graph G, it holds that γc (G)/γ(G) 6
2. This bound is attained by (P7 , C7 )-free graphs with arbitrarily large values
of γ. In particular, the bound γc (G) 6 2γ(G) is the best possible one even in
the class of (P7 , C7 )-free graphs.
Proof. Let G = (V, E) be a (P8 , C8 )-free graph. As described before, we can
assume that G is connected. Let D be a minimum dominating set of G. Let
D1 , D2 , . . . , Dk be the vertex sets of the connected components of G[D].
It is clear that if k = 1, then γc (G) = γ(G). So assume that G[D] has at
least two connected components, which is, k > 2. Let C ⊆ V be an inclusionwise minimal set such that G[D ∪ C] is connected, and let X ⊆ D ∪ C be
a minimal connected dominating set of G. By Lemma 5.5, G[X] is P6 -free.
The situation is illustrated by Figure 5.1.
Let us first assume that γ(G) = |D| 6 3. If k = 2, then γc (G) 6 γ(G)+2,
and since γc (G) > γ(G) > 2, this implies that γc (G)/γ(G) 6 2. So we may
assume that k = 3. Then D is an independent set of size 3. Let D = {x, y, z}.
Since D is a dominating set of G and {x}, {y}, {z} are the vertex set of
connected components in G[D], there exists a nearest connected component,
say {y}, of G[D] such that the distance from {x} is at most 3. Similarly,
there exists a nearest connected component, say {y}, of G[D] such that
the distance from {z} is at most 3. So, at most four vertices of C suffice
to connect x, y and z. By the minimality of C, |C| 6 4. Thus, |X| 6
|D| + |C| 6 7.
Suppose that |X| = 7, i.e., |C| = 4 and X = D ∪ C. Because |C| = 4,
by the previous argumentation, we can suppose that there are two vertices
u and v in C such that xuvy is a shortest path in G[X] between x and y,
and there are two vertices u0 and v 0 in C such that yv 0 u0 z is a shortest path
in G[X] between y and z. Because |C| = 4, all of u, v, u0 , v 0 are distinct by
minimality of C. Then {u, v, u0 , v 0 } ∩ D = ∅ and C = {u, v, u0 , v 0 }. The
situation is illustrated by Figure 5.3. We will investigate different cases
depending on the adjacency between vertices from {x, y, z, u, u0 , v, v 0 }.
If x, resp. z, is adjacent to at least one of u0 or v 0 , resp. u or v, then by
minimality of C, C = {u0 , v 0 }, resp. C = {u, v}, a contradiction. Otherwise,
58
Chapter 5. The price of connectivity for domination
x
u
v
v0
y
u0
z
Figure 5.3: The situation when D = {x, y, z} and C = {u, u0 , v, v 0 }.
if u, resp. u0 , is adjacent to at least one of u0 or v 0 , resp. u or v, then
by minimality of C, C = {u, u0 , v 0 }, resp. C = {u0 , u, v}, a contradiction.
Otherwise G[X] is isomorphic to P7 , or contains an induced P6 , depending
on the adjacency between v and v 0 , a contradiction.
This means |X| 6 6 and thus γc (G) 6 |X| 6 6 = 2γ(G) in this case.
Now let us assume that γ(G) = |D| > 4. Let I ⊆ {1, 2, . . . , k} be such
that i ∈ I if and only if Di ∩ X = ∅. For every i ∈ I, pick xi ∈ X such that
xi has a neighbor in Di ; this is always possible, since X is a dominating set
of G. Note that every
S xi belongs to C, and that the xi do not have to be
distinct. Let S = i∈I
/ (Di ∩ X) ∪ {xj | j ∈ I}. To estimate the cardinality
of X, we will prove that X mainly consists of all vertices from S and others
from a connected dominating set of G[X].
It is shown by van ’t Hof and Paulusma [122] that every connected P6 -free
graph has a connected dominating set Z for which the following assertion
holds: G[Z] is either isomorphic to C6 or contains a complete bipartite graph
as spanning subgraph. Let Y be such a connected dominating set of G[X].
Assume first that G[Y ] is isomorphic to C6 . Let u1 u2 · · · u6 be a consecutive ordering of the vertices of the C6 . Suppose that Y 0 = {u1 , u2 , u3 , u4 }
is not a dominating set of G[X]. Then there is a vertex z ∈ X with
NG [z] ∩ Y 0 = ∅. Without loss of generality, z is adjacent to u5 . But then, as
illustrated by Figure 5.4, G[Y 0 ∪ {u5 , z}] is isomorphic to P6 , a contradiction
to the fact that G[X] is P6 -free.
Thus Y 0 is a connected dominating set of G[X].
Since {xj | j ∈ I} ⊆ X and Y 0 is a connected dominating set of G[X],
G[Y 0 ∪{xj | j ∈ I}] is connected. Thus, G[D ∪Y 0 ∪{xj | j ∈ I}] is connected:
every Di with Di ∩X 6= ∅ has a neighbor in Y 0 , and every Di with Di ∩X = ∅
has a neighbor in {xj | j ∈ I}, namely xi .
As (Y 0 ∪ {xj | j ∈ I}) ⊆ X, (Y 0 ∪ {xj | j ∈ I}) \ D ⊆ C. By the
minimality of C, C = (Y 0 ∪ {xj | j ∈ I}) \ D. Thus, X ⊆ S ∪ Y 0 , which gives
X
γc (G) 6 |X| 6 |S| + |Y 0 | 6
|Di ∩ X| + |I| + 4 6 |D| + 4 6 2γ(G).
i∈I
/
Now assume that G[Y ] contains a complete bipartite graph as a spanning
subgraph. Let (A, B) be a bipartition of this complete bipartite graph. For
59
5.2 Structural results
u3
u2
u4
u1
u5
u6
z
G[X]
Figure 5.4: G[Y ] is isomorphic to C6 , where Y = {u1 , u2 , u3 , u4 , u5 , u6 } is a
dominating set of G[X], and {u1 , u2 , u3 , u4 } is not a dominating set of G[X].
each 1 6 i 6 k, pick yi ∈ Y with the following property. If Di ∩ X 6= ∅,
NG[X] [yi ] ∩ (Di ∩ X) 6= ∅, and if Di ∩ X = ∅, yi ∈ NG[X] [xi ]. The situation is
illustrated by Figure 5.5. These yi exist since Y is a dominating set of G[X].
D1
y1
x1
y4
D4
y2
D2
x3
y3
x5
D5
A
B
G[Y ]
D3
G[X]
Figure 5.5: G[Y ] contains a complete bipartite graph as a spanning subgraph,
where x5 = y5 .
We can assume that A ∩ {yi | 1 6 i 6 k} =
6 ∅.
If B ∩ {yi | 1 6 i 6 k} =
6 ∅, the graph G[D ∪ {xi | i ∈ I} ∪ {yj |
1 6 j 6 k}] is connected. As {xi | i ∈ I} ∪ {yj | 1 6 j 6 k} ⊆ X,
({xi | i ∈ I} ∪ {yj | 1 6 j 6 k}) \ D ⊆ C. By the minimality of C,
C = ({xi | i ∈ I} ∪ {yj | 1 6 j 6 k}) \ D. Thus X ⊆ S ∪ {yj | 1 6 j 6 k},
which gives
X
γc (G) 6 |X| 6 |S| + k 6
|Di ∩ X| + |I| + k 6 |D| + k 6 2γ(G). (5.3)
i∈I
/
So we may assume that B ∩ {yi | 1 6 i 6 k} = ∅, as illustrated by
Figure 5.6.
60
Chapter 5. The price of connectivity for domination
D1
x1
D4
y1
y2
D2
x3
D3
y3
x5
D5
z
B
A
G[Y ]
G[X]
Figure 5.6: G[Y ] contains a complete bipartite graph as a spanning subgraph,
where B ∩ {yi | 1 6 i 6 k} = ∅ and for instance, x5 = y5 , y2 = y4 .
Pick any z ∈ B. Since D is a dominating set of G, there is an index
1 6 l 6 k such that NG [z] ∩ Dl 6= ∅. In the Figure 5.6, l = 5. Hence,
G[D ∪ {xi | i ∈ I} ∪ {yj | 1 6 j 6 k, j 6= l} ∪ {z}] is connected. So,
C = ({xi | i ∈ I} ∪ {yj | 1 6 j 6 k, j 6= l} ∪ {z}) \ D and thus X ⊆ S ∪ {yj |
1 6 j 6 k, j 6= l} ∪ {z}, which gives (5.3). This completes the proof of the
bound γc (G) 6 2γ(G) for (P8 , C8 )-free graphs.
The bound γc (G) 6 2γ(G) is attained by an infinite number of connected
(P7 , C7 )-free graphs G, given by the following construction. For every k ∈ N,
let Hk be the graph obtained from Kk by attaching to each vertex a pendent
path on two vertices in the way illustrated in Figure 5.7. It is easily seen
that, for all k ∈ N, γc (Hk )/γ(Hk ) = 2.
Figure 5.7: The set of black vertices indicates a minimum dominating set
(respectively a minimum connected dominating set) of H4 .
A similar construction shows that the inequation (5.1) in page 49 is
asymptotically sharp in the class of (P9 , C9 )-free graphs, in the sense that
there is an infinite family {Gk | k ∈ N} of (P9 , C9 )-free graphs such that
limk→∞ γc (Gk )/γ(Gk ) = 3. For every k ∈ N, let Gk be the graph obtained
from Kk by attaching to each vertex a pendent path on three vertices in
5.2 Structural results
61
the way illustrated in Figure 5.8. It is easy to check that for every k > 2,
γ(Gk ) = k + 1 and γc (Gk ) = 3k. Furthermore, Gk is (P9 , C9 )-free.
Figure 5.8: The set of black vertices indicates a minimum dominating set
(respectively a minimum connected dominating set) of G4 .
We end this section with a conjecture that came up during our research.
As Theorem 5.8 shows, it holds that γc (G) 6 2γ(G) for every (P8 , C8 )-free
graph G. However, γc (P8 )/γ(P8 ) = 2 = γc (C8 )/γ(C8 ), i.e., both P8 and C8
do not violate the bound given by Theorem 5.8. While intensively searching
for minimal connected graphs G with γc (G) > 2γ(G), we had the strong
impression that P9 , C9 and H, the graph depicted in Figure 5.9, might be
the only minimal graphs. If this is true, the following conjecture holds.
Conjecture 5.9. For every (P9 , C9 , H)-free graph G, where H is the graph
depicted in Figure 5.9, it holds that γc (G) 6 2γ(G).
Note that for any G ∈ {P9 , C9 , H}, γc (G) > 2γ(G). Hence, if true,
Conjecture 5.9 would give a characterization of the largest class of connected
graphs that is closed under connected induced subgraphs where γc (G) 6
2γ(G) holds, i.e. would give an exact characterization of PoC-near-perfect
graphs with threshold 2.
Figure 5.9: The graph H from Conjecture 5.9.
5.2.2
PoC-Critical Graphs
We now turn our attention to critical graphs, which is, graphs G for which
the price of connectivity of every proper induced subgraph H of G is strictly
smaller than the price of connectivity of G, similarly to what we did for
the vertex cover problem. These are exactly the graphs that can appear
in a minimal forbidden induced subgraph characterization of the PoC-nearperfect graphs for some threshold r ∈ [1, 3). A perhaps more tractable class
of graphs is the class of strongly critical graphs, defined as the graphs G for
62
Chapter 5. The price of connectivity for domination
which every proper (not necessarily induced) subgraph H of G has a price
of connectivity that is strictly smaller than the price of connectivity of G. It
is clear that every strongly critical graph is critical, but the converse is not
true. For instance, C5 is critical, but not strongly critical. Notice that every
(strongly) critical graph is connected.
PoC-Critical Trees: Let T be a tree. We call T special if T is obtained
from another tree (with at least one edge) by subdividing each edge either
once or twice and then attaching a pendent vertex to every leaf of the resulting graph (see Figure 5.10 for an example).
Figure 5.10: A special tree constructed from another tree (vertices indicated
by filled circles) by subdividing each edge either once or twice (subdivision
vertices are indicated by hollow circles) and then attaching a pendent vertex
(indicated by squares) to every leaf.
The next result gives a partial characterization of the class of critical
trees. However, the class of special trees turns out to be too restricted. We
need a new definition.
We call a tree T peculiar if
• the neighbor of every leaf has degree 2,
• every minimum dominating set D of T is independent and
• every vertex v ∈ V (T ) \ D with degree at least 3 has only one neighbor
in D, i.e. |NT (v) ∩ D| = 1.
See Figure 5.11 for an example.
Figure 5.11: A peculiar tree where a minimum dominating set contains vertices indicated by filled circles and leaves are indicated by squares.
5.2 Structural results
63
In spite of our initial expectation, Theorem 4.12 cannot be straightforwardly adapted to the domination case because special trees (in the sense of
this section) are too restricted.
Theorem 5.10. For a tree G, the following assertions are equivalent:
(i) G is a peculiar critical tree.
(ii) G is strongly critical.
(iii) G is critical.
Moreover, if G is critical and if the degree of any v ∈ V (G) \ D, where D is
an arbitrary minimum dominating set of G, is at most 2, then G is a special
tree built on an initial tree H, where V (H) is a minimum dominating set.
Before proving the theorem, we show the following useful lemma.
Lemma 5.11. Let G be a critical graph. For every minimum dominating
set D of G, there does not exist a bridge of G with endpoints in D.
Proof. Suppose there exists a bridge x1 x2 with x1 , x2 ∈ D. The removal of
the edge x1 x2 results in two connected subgraphs of G, which we denote by
G1 and G2 respectively. We can assume that x1 ∈ V (G1 ) and x2 ∈ V (G2 ).
Let G01 be the graph obtained from G1 by attaching a pendent vertex to
x1 . Similarly, let G02 be the graph obtained from G2 by attaching a pendent
vertex to x2 .
We observe that D ∩ V (G1 ) is a dominating set of G01 and D ∩ V (G2 ) is
a dominating set of G02 . Thus
γ(G) > γ(G01 ) + γ(G02 ).
(5.4)
On the other hand, let Dc,1 be a minimum connected dominating set of
G01 and Dc,2 be a minimum connected dominating set of G02 . We can assume
that Dc,1 ⊆ V (G1 ) and Dc,2 ⊆ V (G2 ), since for i = 1, 2, G0i is the graph
Gi with a pendent vertex. It is clear that x1 ∈ Dc,1 and x2 ∈ Dc,2 . Thus
Dc,1 ∪ Dc,2 is a connected dominating set of G. Since Dc,1 ∩ Dc,2 = ∅,
γc (G) 6 γc (G01 ) + γc (G02 ).
(5.5)
But (5.4) and (5.5) say that
γc (G)/γ(G) 6 max{γc (G01 )/γ(G01 ), γc (G02 )/γ(G02 )}.
(5.6)
Since both G01 and G02 are isomorphic to induced subgraphs of G, (5.6) is a
contradiction to the fact that G is critical.
64
Chapter 5. The price of connectivity for domination
Proof of Theorem 5.10. It is obvious that (ii) implies (iii). Firstly, we show
that (i) implies (ii), which is, every peculiar critical tree is strongly critical.
Let G be a peculiar critical tree. Let G0 be a proper (not necessarily induced)
subgraph of G and C1 , . . . , Ck be the vertex sets of its connected components.
Because G is a tree, G0 is the disjoint union of G[Ci ] for i = 1, . . . , k. Moreover a minimum (resp. connected) dominating set of G0 is the union of a
minimum (resp. connected) dominating set of each connected component of
G’, so
P
γc (G0 )
γc (G[Ci ]) γc (G)
16i6k γc (G[Ci ])
P
6 max
=
i = 1, . . . , k <
,
γ(G[C
])
γ(G0 )
γ(G[C
])
γ(G)
i
i
16i6k
since G is critical.
Secondly, we prove that (iii) implies (i), which is, every critical tree is
peculiar. For this, let G = (V, E) be a critical tree. Let D be a minimum
dominating set of G.
First we show that D is an independent set. Suppose there are x, y ∈ D
such that xy ∈ E. Since G is a tree, xy is a bridge, a contradiction to
Lemma 5.11.
Now we show that every member of V \ D with degree at least 3 has only
one neighbor in D. For this, let x ∈ V \ D with |NG (x)| > 3. Suppose that
|NG (x) ∩ D| > 2, hence let d1 , d2 ∈ NG (x) ∩ D. Let X1 , X2 , . . . , Xk be the
vertex sets of the connected components of G − x. By assumption, k > 3.
Suppose that d1 ∈ X1 and d2 ∈ X2 . Let
[
H1 = G −
Xi
i6=2,3
and
H2 = G − (X2 ∪ X3 ).
We observe that
γ(G) > γ(H1 ) + γ(H2 ).
(5.7)
Since x is a cutvertex of H1 , x is contained in every connected dominating
set of H1 . Therefore
γc (G) 6 γc (H1 ) + γc (H2 ).
(5.8)
By the same argumentation from Lemma 5.11, (5.7) and (5.8) yield a
contradiction to the fact that G is critical. This proves that every vertex of
V \ D with degree at least 3 has only one neighbor in D. From the discussion
above, D is an independent set.
Moreover, two degree-1 vertices, say x and y, cannot have the same
neighbor since γc (G − x) = γc (G), γ(G − x) = γ(G) and G is critical.
The neighbor y of any degree-1 vertex x must have its degree equal to 2.
Otherwise, clearly γ(G − x) 6 γ(G). The vertex y is a cutvertex of the
65
5.2 Structural results
tree G − x, hence γc (G − x) = γc (G). We obtain a contradiction since G is
critical. All in all, G is peculiar.
Now, we prove that the critical tree G is special if every member of V \ D
has a degree at most 2, for a minimum dominating set D. By the previous
argument, we know that any minimum dominating set is independent. We
can suppose that D does not contain a leaf. Otherwise, we could replace
in D the degree-1 vertex by its neighbor such that D is an independent
dominating set and every vertex v ∈
/ D has degree at most 2. Consider the
initial tree H defined as followed: V (H) = D and E(H) = {uv| there exists
a path Puv in (V \ D) ∪ {u, v} from u to v}. Because D is a dominating set
of G, if uv is an edge in H, then the length of the path Puv in G is either 2
or 3. All in all, G is a special tree. This completes the proof.
Notice that Theorem 5.10 is quite similar to Theorem 4.12 for the vertex cover problem but the hypothesis in the domination version is stronger.
Indeed, for the dominating set problem, in the class of chordal graphs, the
three assumptions from Theorem 5.10 are not equivalent: the graph from
Conjecture 5.9 is a critical chordal graph without being a tree.
Now, we investigate the relations between graph classes: special trees,
peculiar trees and critical trees. Figure 5.12 illustrates the situation.
Figure 5.11
P8
Figure 5.14
Figure 5.13
P7
critical trees
special trees
peculiar trees
Figure 5.12: The situation around critical trees, where the gray area represents special trees with a double subdivision.
Not all peculiar trees are critical. For instance, the graph depicted in
Figure 5.11 (whose price of connectivity is 12/5) is not critical because it
contains an induced subgraph H with a higher price of connectivity. Indeed,
for instance, H could be the graph obtained from K1,3 by subdividing each
edge exactly thrice. Furthermore, the graph illustrated by Figure 5.13 is a
peculiar critical tree which is not special. Also, we point out that not all
66
Chapter 5. The price of connectivity for domination
special trees are critical, for instance P8 contains an induced P6 with the
same price of connectivity.
Figure 5.13: A peculiar critical tree which is not special, where a minimum
dominating set contains vertices indicated by filled circles and leaves are
indicated by squares.
Moreover, by Proposition 5.12, every special tree built on the initial
tree H, where all edges of H are subdivided exactly twice in G, is critical.
These graphs are represented by the gray area in Figure 5.12. However, the
converse in the class of special trees is not true because the graph illustrated
by Figure 5.14 is critical.
Proposition 5.12. Let G be a special tree built on the initial tree H. If all
edges of H are subdivided twice in G, then G is critical.
Proof. Notice that γc (G) = 3γ(G) − 2. Let G0 be a proper induced subgraph
of G.
First, we suppose that γ(G0 ) = γ(G). We know that a minimum connected dominating set of a tree (with one vertex of degree > 1) is the set
of vertices with degree at least 2, i.e. all vertices which are not a leaf, or in
other words, the set of internal vertices. Observe that the number of internal
vertices of G0 is strictly smaller than this of G because every neighbor of a
leaf in G has a degree 2. Thus γc (G0 ) < γc (G) and
γc (G0 )
γc (G)
<
.
0
γ(G )
γ(G)
Now, assume that γ(G0 ) > γ(G). Because G is a tree, a minimum connected dominating set of G0 is a subset of the internal vertex set of G, hence
γc (G0 ) 6 γc (G). Trivially, we obtain
γc (G)
γc (G0 )
>
.
γ(G)
γ(G0 )
It remains the case where γ(G0 ) < γ(G). Since γc (G0 ) 6 3γ(G0 ) − 2,
γc (G0 )
3γ(G0 ) − 2
2
2
3γ(G) − 2
γc (G)
6
=3−
<3−
=
=
.
0
0
0
γ(G )
γ(G )
γ(G )
γ(G)
γ(G)
γ(G)
5.2 Structural results
67
Figure 5.14: A special critical tree constructed from another tree H (vertices
indicated by filled circles) where each edge of H is not necessarily subdivided
twice.
Besides, the following proposition gives a necessary condition for a special
tree to be critical. First, we prove a short lemma.
Lemma 5.13. Every special tree G built on an initial tree H with |V (H)| > 3
contains a proper induced path on 6 vertices.
Proof. Let P be a longest path in H. Then P has k > 3 vertices, since
|V (H)| > 3. Thus, the corresponding path P 0 in G contains all vertices
from P and at least one for each edge. Moreover, P 0 can be extended in its
endpoints by two vertices, by construction of G. Accordingly, G contains
a path on |V (P 0 )| + 2 > (2k − 1) + 2 > 7 vertices. Besides, being a tree
implies that any path is induced. Thus, G contains a proper induced path
on 6 vertices.
Proposition 5.14. Let G be a special critical tree built on the initial tree H
with V (H) a minimum dominating set. Suppose that |V (H)| > 3.
(i) Let x be a leaf of H and y be its neighbor in H. Then the edge xy is
subdivided exactly twice in G.
(ii) Let v be a vertex in H and NH (v) be its neighborhood. Then there exists
at least one u ∈ NH (v) such that the edge uv is subdivided exactly twice
in G.
Proof. By Lemma 5.13, observe that γc (P6 )/γ(P6 ) = 2 < γc (G)/γ(G) since
G is critical.
First, we prove (i). Suppose that there exists a leaf x in H and its
neighbor y where xy is subdivided exactly once in G. Let x0 be the pendent
vertex of x in G. Consider G0 = G − {x, x0 } the induced subtree built on
H − {x}. It is easy to see that γ(G) > γ(G0 ) + 1 and γc (G) 6 γc (G0 ) + 2.
Thus,
γc (G)
γc (G0 ) + 2
γc (G0 )
γc (G0 )
6
6
max
,
2
=
.
(5.9)
γ(G)
γ(G0 ) + 1
γ(G0 )
γ(G0 )
The last equality is true because γc (G)/γ(G) > 2. Thus, we obtain a
contradiction since G is critical.
It remains to show (ii). Let v be a vertex in H. Suppose that for every
u ∈ NH (v), the edge uv is subdivided only once in G. We can suppose that
68
Chapter 5. The price of connectivity for domination
dH (v) > 2 because the case of leaves was studied previously. Let u1 , . . . , uk
be the neighbors of v in H with k > 2 and vi be the midpoint of the edge
ui v in G, for all i. Consider C1 , . . . , Cj the vertex sets of the connected
components of G − v. Since G is a tree, we have that j = k and we can
assume that vi , ui ∈ Ci for all i.
Let G1 be the subgraph of G induced by C1 and G2 be the subgraph of G
induced by ∪ki=2 Ci ∪ {v, v1 , u1 }. We observe that V (H) ∩ C1 is a dominating
set of G1 and ((V (H) ∩ V (G2 )) \ {v}) ∪ {v1 } is a dominating set of G2 . Thus
γ(G) > γ(G1 ) + γ(G2 ).
(5.10)
On the other hand, let Dc,1 be a minimum connected dominating set of
G1 and Dc,2 be a minimum connected dominating set of G2 . We can assume
that Dc,1 ⊆ V (G1 ) \ {v1 } and Dc,2 ⊆ V (G2 ) \ {u1 }. It is clear that u1 ∈ Dc,1
and v1 ∈ Dc,2 . Thus Dc,1 ∪ Dc,2 is a connected dominating set of G. Since
Dc,1 ∩ Dc,2 = ∅,
γc (G) 6 γc (G1 ) + γc (G2 ).
(5.11)
But (5.10) and (5.11) say that
γc (G)/γ(G) 6 max{γc (G1 )/γ(G1 ), γc (G2 )/γ(G2 )}.
(5.12)
Since both G1 and G2 are induced subgraphs of G, (5.12) is a contradiction
to the fact that G is critical.
PoC-Strongly-Critical Graphs: The following result is similar to Theorem 4.14, except that in the case of the dominating set problem, there exist
strongly critical graphs that are not trees but have a cutvertex (see the graph
in Figure 5.9).
Proposition 5.15. Let G be a strongly critical graph. Every minimum dominating set of G is independent.
Proof. Let G be a strongly critical graph and let D be a minimum dominating
set of G. Suppose that D is not an independent set. Thus there are two
adjacent vertices in D, say x and y.
By Lemma 5.11, xy cannot be a bridge of G. So G − xy is connected.
Clearly γ(G) > γ(G − xy). Every minimum connected dominating set of
G − xy is trivially a connected dominating set of G. Hence γc (G) 6 γc (G −
xy). Therefore,
γc (G − xy)
γc (G)
>
,
γ(G − xy)
γ(G)
which is a contradiction.
5.2 Structural results
5.2.3
69
Aside
In this section, we investigate the price of connectivity in relationship with
other invariants such as the radius and the diameter. This is similar in spirit
to [30]. Moreover, we prove that for all rationals r ∈ [1, 3), there exists a
graph whose price of connectivity is exactly r.
Price of connectivity and radius
The following result describes the relationship between the price of connectivity and radius.
Theorem 5.16. Let G be a graph.
• If the radius of G is 1, then γc (G)/γ(G) = 1.
• If the radius of G is 2, then γc (G)/γ(G) 6 2.
• For every fixed integer k > 3, if the radius of G is k, then γc (G)/γ(G) <
3.
Moreover, all the bounds are (asymptotically) tight, even for graphs G with
large γ(G).
Proof. If the radius of G is 1, then there exists one vertex adjacent to all
the others. In this case, only this vertex is sufficient to dominate the whole
graph, which means γc (G)/γ(G) = 1.
Suppose that the radius of G is exactly 2. Let c be a center of the
graph. For every vertex v in G, the distance between c and v is at most 2.
Let D be a minimum dominating set of G with the connected components
D1 , . . . , Dr , where D1 is a nearest connected components of D to c (it could
be c ∈ D1 ). Observe that d(D1 , c) is at most 1, because D is a dominating
set. We assume that r > 1. Note that, for every i > 2, the distance between
Di and c is at most 2, i.e. either Di is adjacent to c or only one vertex, say
xi , is sufficient to connect Di to c. We define I = {i|d(Di , c) = 2}. In this
case, D ∪ {c} ∪ {xi |i ∈ I} is a connected dominating set by construction.
Furthermore,
γc (G)
|D| + 1 + |{xi |i ∈ I}|
6
6 2.
γ(G)
|D|
We build a tight graph H˜n . Take a clique on n vertices, one of which is
the center c. Choose another vertex v from the clique and add a pendent
vertex to it. Then, for every vertex u in the clique different from c and v,
add a cycle of length 5 through c and u. See Figure 5.15. Since the set
containing the vertex v and the vertex which is at distance 2 from the clique
for each 5-vertex cycle is a minimum dominating set, γ(H˜n ) = n − 1. As
illustrated by Figure 5.15, γc (H˜n ) = 2n − 2.
70
Chapter 5. The price of connectivity for domination
c
c
v
v
Figure 5.15: The set of black vertices indicates a minimum dominating set
(respectively a minimum connected dominating set) of H̃4 with center c.
Let k > 3 be a fixed integer. For all graphs G, we have γc (G)/γ(G) < 3.
To see that this bound is the best possible one for graphs with radius k,
consider the infinite family {Hn,k |n ∈ N} of graphs with radius k, where
Hn,k is a flower with n cycles of size 6 and one of size 2k. An illustration of
H3,k is given by Figure 5.16. Clearly γ(Hn,k ) > n + 2k
3 . Since the set of all
vertices except two vertices for each induced cycle is a connected dominating
set, γc (Hn,k ) 6 3n + 2k − 2. Hence for any fixed k,
γc (Hn,k )
3n + 2k − 2
6
−→ 3.
γ(Hn,k )
n + (2k)/3 n→+∞
...
...
...
...
Figure 5.16: The set of black vertices indicates a minimum dominating set
(respectively a minimum connected dominating set) of H3,k .
Price of connectivity and diameter
Before establishing result on graphs with diameter 2, we state the following
useful lemma.
71
5.2 Structural results
Lemma 5.17. Let G be a graph on n > ∆2 − ∆ − 1 vertices of diameter 2
with ∆(G) = ∆. If there exists one vertex v with dG (v) < ∆ then
γc (G)/γ(G) 6 3/2.
Proof. We have
|V (G)|
∆+1
∆2 − ∆ − 1
>
∆+1
1
=∆−2+
.
∆+1
γ(G) >
Hence, γ(G) > ∆ − 1. Moreover, a connected dominating set is N [v], so
γc (G) 6 ∆. Thus, γc (G)/γ(G) 6 ∆/(∆ − 1) 6 3/2 if and only if ∆ > 3. If
∆ 6 2 then G is a path on k 6 3 vertices or a cycle on r 6 5 vertices, since
G is connected. In each case, its price of connectivity is at most 3/2.
Bondy, Erdős and Fajtlowicz [23] established that every graph G of order
n with diameter 2 and no cycle of length 4 satisfies that either
- ∆(G) = n − 1, or
- G is a Moore graph, i.e. d-regular with 1 + d2 vertices, or
- ∆(G) = ∆, |V (G)| = ∆2 − ∆ + 1 and each vertex of G has degree ∆
or ∆ − 1.
Theorem 5.18. Let G be a graph of diameter 2 with no cycle of length 4.
Then γc (G)/γ(G) = 3/2.
Proof. Obviously, if ∆(G) = n − 1 then γc (G) = γ(G) = 1. Suppose that G
is a Moore graph. We have
|V (G)|
γ(G) >
∆(G) + 1
1 + d2
=
1+d
2
d −1
>
1+d
= dd − 1e
= d − 1.
Hence γ(G) > d. Besides, a connected dominating set is N [v] for any v, so
γc (G) 6 d + 1. Thus, γc (G)/γ(G) 6 (d + 1)/d 6 3/2 since d > 2.
72
Chapter 5. The price of connectivity for domination
Now, assume that |V (G)| = ∆2 − ∆ + 1 and each vertex of G has degree ∆ or ∆ − 1. If there is a vertex v such that d(v) = ∆ − 1, then
Lemma 5.17 gives immediately the result. Otherwise, G is ∆-regular. By a
similar argumentation in Lemma 5.17, we have that γ(G) > ∆ − 1. Moreover, a connected dominating set is N [v] for any v, so γc (G) 6 ∆ + 1. Thus,
γc (G)/γ(G) 6 (∆ + 1)/(∆ − 1) 6 3/2 if and only if ∆ > 5. Besides, two
systems for computer aided graph theory [39, 40] and a graph generator [96]
verify the upper bound on the price of connectivity for all ∆-regular graphs
on at most 13 vertices with diameter 2, completing the proof.
Values of the price of connectivity
Theorem 5.19. Let r be a rational number in [1, 3). There exists a (P9 , C9 )free graph G such that γc (G)/γ(G) = r.
Proof. Assume that r = pq , with p and q two integers. Let s = p − q.
Consider Fp,q the graph obtained from a clique K2q on 2q vertices by adding
to s vertices of the clique a pendent path on three new vertices and a pendent
vertex to the other vertices, as illustrated by Figure 5.17. Clearly, Fp,q is
(P9 , C9 )-free.
Figure 5.17: The set of black vertices indicates a minimum dominating set
(respectively a minimum connected dominating set) of F5,2 .
We will prove that γ(Fp,q ) = 2q and γc (Fp,q ) = 2p. Note that s < 2q since
r < 3. Hence, at least one vertex from the clique is attached to a pendent
vertex. A minimum dominating set consists of the middle vertex from each
pendent path of length 3 and the vertex in the clique for each pendent vertex,
as illustrated by Figure 5.17, which implies that γ(Fp,q ) = 2q. Furthermore,
a minimum connected dominating set is the set of all vertices with degree
> 1, which means that γc (Fp,q ) = 3s + (2q − s) = 2p.
Chapter 6
A characterization of Pk -free
graphs
The class of graphs that do not contain an induced path on k vertices, Pk -free
graphs, plays a prominent role in algorithmic graph theory. Let us mention
a recent result: there exists a polynomial-time algorithm to compute an independent set of maximum weight in a P5 -free graph, given by Lokshtanov,
Vatshelle and Villanger [93]. This prominent role motivates the search for
special structural properties of Pk -free graphs, including alternative characterizations. Bácso and Tuza are the precursors in the study of Pk -free graphs
with the following theorem.
Theorem 6.1 (Bácso and Tuza [11]). Let G be a connected P5 -free graph.
Then G has a dominating clique or a dominating induced P3 .
An immediate consequence of this result is the following theorem.
Theorem 6.2 (Bácso and Tuza [11], Cozzens and Kelleher [49]). Let G be
a graph. The following assertions are equivalent.
(i) G is P5 -free.
(ii) Every connected induced subgraph H of G admits a connected dominating set X such that H[X] is a clique or H[X] ∼
= C5 .
Later, van ’t Hof and Paulusma [122] obtained a characterization for the
class of P6 -free graphs in the flavor of Theorem 6.2. An earlier, slightly
weaker result was given by Liu, Peng and Zhao [91], and the particular case
of (C3 , P6 )-free graphs was discussed before by Liu and Zhou [92].
Theorem 6.3 (van ’t Hof and Paulusma [122]). Let G be a graph. The
following assertions are equivalent.
(i) G is P6 -free.
73
74
Chapter 6. A characterization of Pk -free graphs
(ii) Every connected induced subgraph H of G admits a connected dominating set X such that H[X] has a complete bipartite spanning subgraph
or H[X] ∼
= C6 .
Complementing Theorem 6.3, van ’t Hof and Paulusma give a polynomial-time algorithm that, given a connected P6 -free graph, computes a connected dominating set X such that G[X] has a complete bipartite spanning
subgraph or G[X] ∼
= C6 .
In view of Theorems 6.2 and 6.3, two questions arise.
• The first one is whether assertion (ii) of Theorem 6.3 can be strengthened, such as H[X] is a P4 -free graph or H[X] ∼
= C6 . Note that if H[X]
is P4 -free, it is a connected cograph, and in particular has a complete
bipartite spanning subgraph. This condition is the direct analogy of
condition (ii) of Theorem 6.2 for P6 -free graphs. The advantage of the
strenghtened version is, of course, that the structure of cographs is well
understood and more restricted compared to the class of graphs having
a spanning complete bipartite graph.
• The second question is whether similar characterizations can be given
for the class of Pk -free graphs, for k > 6. In their paper, van ’t Hof and
Paulusma [122] explicitly ask for such a characterization in the case of
k = 7.
This chapter and the next one are based on a joint paper published first in
the 40th International Workshop on Graph-Theoretic Concepts in Computer
Science (WG2014) and then in Algorithmica [34]. The paper received a best
paper award in WG2014. This chapter is split into two sections.
• In the first section, we give an affirmative answer to the two questions
above. We show that every Pk -free graph, k ≥ 4, admits a connected
dominating set that induces either a Pk−2 -free graph or a graph isomorphic to Pk−2 , hence generalizing Theorem 6.1. Surprisingly, it turns
out that every minimum connected dominating set of G has this property. This yields a new characterization for Pk -free graphs, which is
similar to Theorem 6.2: a graph G is Pk -free if and only if each connected induced subgraph of G has a connected dominating set that
induces either a Pk−2 -free graph, or a graph isomorphic to Ck . Notice that Pk−2 at the end of the first assertion was changed to Ck in
order to handle the case of Pk . Accordingly, our results strengthen
Theorem 6.3.
• In the second section, we present an efficient algorithm that, given
a connected graph G, computes a connected dominating set X of G
with the following property based on our first result: for the minimum
k such that G is Pk -free, the subgraph induced by X is Pk−2 -free or
6.1 A characterization of Pk -free graphs
75
isomorphic to Pk−2 . A property of our algorithm is that it does not
require the value k, which is N P -complete to compute.
Our theorems have other applications. One example we will give in the
next chapter is Hypergraph 2-Colorability for hypergraphs with P7 free incidence graphs.
Other possible future applications of our results include the coloring of
Pk -free graphs. Hoang, Kamiński, Lozin, Sawada and Shu [77] showed that kColorability is efficiently solvable on P5 -free graphs, using the fact that a
connected P5 -free graph has a dominating clique or a dominating induced P3 .
As a private communication, Bonomo, Chudnovsky, Maceli, Schaudt, Stein
and Zhong [24] use our characterization for proving that 3-Colorability
is polytime-time solvable on P7 -free graphs, which will be an important application of our results. They prove that a 3-colorable P7 -free graph has a
set X of at most three vertices such that the distance from any vertex to X
is at most 2, by applying twice our results.
6.1
A characterization of Pk -free graphs
As mentioned previously, we show that every connected Pk -free graph, k ≥ 4,
admits a connected dominating set that induces either a Pk−2 -free graph, or
a graph isomorphic to Pk−2 . Surprisingly, it turns out that every minimum
connected dominating set has this property.
Theorem 6.4. Let G be a connected Pk -free graph, k ≥ 4, and let X be
any minimum connected dominating set of G. Then G[X] is Pk−2 -free, or
G[X] ∼
= Pk−2 .
Let us recall the following lemma, proved in Chapter 5.
Lemma 6.5 (Camby and Schaudt [36]). Let G be a connected graph that is
(Pk , Ck )-free, for some k ≥ 4, and let X be a minimal connected dominating
set of G. Then G[X] is Pk−2 -free.
When applied to connected Pk -free graphs, which are in particular (Pk+1 ,
Ck+1 )-free, the above lemma implies that any minimal connected dominating
set induces a Pk−1 -free graph, for k ≥ 3.
Let X be a connected dominating set of a connected graph G, and x ∈ X.
Assuming that X is a minimal connected dominating set and |X| ≥ 2, x is a
cutvertex of G[X] or x has a private neighbor : a vertex y ∈ V (G) \ X with
NG (y) ∩ X = {x}.
Next we prove a simple but useful lemma, which plays a key role also in
the proof of Theorem 6.8, for our algorithm in the next section.
Lemma 6.6. Let G be a connected Pk -free graph, for some k ≥ 4, and
let X be a minimal connected dominating set of G. Assume that there is an
76
Chapter 6. A characterization of Pk -free graphs
induced Pk−2 in G[X], say on the vertices x1 , x2 , . . . , xk−2 . Then any private
neighbor y of x1 is such that (X ∪ {y}) \ {xk−2 } is a connected dominating
set of G.
Proof. Let X 0 := {x1 , x2 , . . . , xk−2 }. Suppose x1 has a private neighbor y,
and let Y := (X ∪ {y}) \ {xk−2 }. We have to prove that Y is a connected
dominating set of G.
Suppose for a contradiction that G[Y ] is not connected. Hence, xk−2
is a cutvertex of G[X]. In particular, there is a vertex t ∈ X such that
NG (t) ∩ X 0 = {xk−2 }. But then G[X 0 ∪ {y, t}] ∼
= Pk , a contradiction.
It remains to show that Y is a dominating set. Suppose the contrary,
that is, there is a vertex z with NG [z] ∩ Y = ∅. As X is a dominating set,
NG [z] ∩ X = {xk−2 }. Because xk−2 is adjacent to Y and z is not adjacent to
Y , z 6= xk−2 . But this means that G[X 0 ∪ {y, z}] ∼
= Pk , a contradiction.
Now we can state the proof of Theorem 6.4.
Proof of Theorem 6.4. Let X be a minimum connected dominating set of G.
Since G is in particular (Pk+1 , Ck+1 )-free, G[X] is Pk−1 -free, by Lemma 6.5.
We have to show that G[X] is Pk−2 -free or isomorphic to Pk−2 .
To validate this, assume there is an induced Pk−2 in G[X], say on the
vertices x1 , x2 , . . . , xk−2 . Let X 0 := {x1 , x2 , . . . , xk−2 }. We will prove that
X \ X 0 = ∅. Note that x1 is not a cutvertex of G[X]: otherwise there is
a vertex x0 ∈ X such that NG (x0 ) ∩ X 0 = {x1 }, and hence G[X 0 ∪ {x0 }] ∼
=
Pk−1 . This is a contradiction. Thus, x1 is not a cutvertex of G[X] and
therefore has a private neighbor with respect to X, say y1 . By Lemma 6.6,
Y1 := (X ∪ {y1 }) \ {xk−2 } is a connected dominating set of G. As X is a
minimum connected dominating set, Y1 is a minimum connected dominating
set, too. Moreover, y1 has no neighbor in X \ {x1 }, in particular in X \ X 0 .
By reapplying the argumentation to Y1 and the induced Pk−2 on y1 , x1 ,
x2 , . . . , xk−3 , we obtain a vertex y2 ∈ V (G) \ Y1 such that Y2 := (Y1 ∪ {y2 }) \
{xk−3 } is a minimum connected dominating set of G and G[Y2 ] contains an
induced Pk−2 on the vertices y2 , y1 , x1 , x2 , . . . , xk−4 . Moreover, y2 has no
neighbor in Y1 \ {y1 }, in particular in X \ X 0 .
Iteratively, we end up with a minimum connected dominating set Yk−2 ,
which is exactly (X \ X 0 ) ∪ {y1 , . . . , yk−2 }. Since, for i = 1, 2, . . . , k − 2, yi
is not adjacent to X \ X 0 and G[Yk−2 ] is connected, X \ X 0 must be empty,
hence X = X 0 . Thus, G[X] = G[X 0 ] ∼
= Pk−2 . This completes the proof.
From Theorem 6.4, we derive the following characterization of Pk -free
graphs.
Theorem 6.7. Let G be a graph and k ≥ 4. The following assertions are
equivalent.
(i) G is Pk -free.
6.2 A polynomial-time algorithm to find a special connected
dominating set in Pk -free graphs
77
(ii) Every connected induced subgraph H of G admits a connected dominating set X such that H[X] is Pk−2 -free or H[X] ∼
= Ck .
Proof. Clearly Pk does not satisfy (ii) for H = Pk . Accordingly, (ii) implies
(i).
Conversely, let H be any connected induced subgraph of G, and let X
be a minimum connected dominating set of H. By Theorem 6.4, H[X] is
Pk−2 -free or H[X] ∼
= Pk−2 . If H[X] is Pk−2 -free, (ii) is satisfied. Otherwise,
let x1 x2 · · · xk−2 be a consecutive ordering of the induced path H[X]. In
particular, x1 and xk−2 are not cutvertices of H[X], since H[X] is Pk−2 . As
X is minimum, there exists a private neighbor yi of xi , for i ∈ {1, k − 2}.
It must be that y1 yk−2 ∈ E(H), since otherwise H[X ∪ {y1 , yk−2 }] ∼
= Pk .
∼
Hence, H[X ∪ {y1 , yk−2 }] = Ck , as desired. So, (i) implies (ii).
6.2
A polynomial-time algorithm to find a special
connected dominating set in Pk -free graphs
Now we give an algorithmic version of Theorem 6.4. Actually, most of the
proof of Theorem 6.4 holds if we start with a minimal connected dominating
set. However, the result would be: for any connected Pk -free graph, there
exists a minimal connected dominating set X of G such that G[X] is either
Pk−2 -free or isomorphic to Pk−2 . Then this proof is constructive in the sense
that it yields an algorithm to compute, given a connected Pk -free graph, a
connected dominating set that induces either a Pk−2 -free graph, or a graph
isomorphic to Pk−2 . This algorithm runs in polynomial time only when k
is fixed, and hence not useful for arbitrary input graphs. However, recall
that the computation of a longest induced path in a graph is an N P -hard
problem. In other words, there is little hope of computing in polynomial
time the minimum k for which the input graph is Pk -free. To overcome
this obstacle, our algorithm can only make implicit use of the absence of an
induced Pk , which is the main difficulty here.
Theorem 6.8. Given a connected graph G with n vertices and m edges, one
can compute in time O(n5 (n + m)) a connected dominating set X with the
following property: for the minimum k ≥ 4 such that G is Pk -free, G[X] is
Pk−2 -free or G[X] ∼
= Pk−2 .
Before we state our algorithm, we need to introduce some notation and
definitions. For this, let us assume we are given a connected input graph G
on n vertices and m edges. Let X be an arbitrary connected dominating set
of G with at least two vertices.
By NC (X) we denote the set of vertices in X that are non-cutting in
G[X], i.e. for every x ∈ NC (X), G[X \ {x}] is connected. Let x be a degree-1
vertex of G[X]. We define the half-path starting in x to be the maximal path
78
Chapter 6. A characterization of Pk -free graphs
xx1 x2 · · · xs in X such that dG[X] (xi ) = 2 for each i ∈ {1, 2, . . . , s − 1}. The
length of the half-path is then s. For example, if the neighbor y ∈ X of x
has degree at least 3, the half-path is simply xy.
To each x ∈ X we assign a weight wX (x) as follows:
1. if dG[X] (x) ≥ 2, put wX (x) = 0, and
2. if dG[X] (x) = 1, put wX (x) = s, where s is the length of the half-path
starting in x.
Finally, the weight w(X) of the set X is given by
X
w(X) =
(wX (x))2 .
x∈X
See Figure 6.1 for an illustration of these definitions.
0
2
0
0
0
0
0
1
Figure 6.1: A graph G. The black vertices form a connected dominating set
X of G, with weights wX as shown. We have w(X) = 22 + 12 = 5.
Let X be the family of all connected dominating sets of G. We next
define a strict partial order ≺ on X as follows. For any two sets X, Y ∈ X ,
we say that X is strictly smaller than Y according to ≺, denoted by X ≺ Y ,
if
1. |X| > |Y |, or
2. |X| = |Y | and w(X) < w(Y ).
We now consider the strict poset (X , ≺). The height of (X , ≺) is the maximum order of a set of mutually comparable elements of X .
Lemma 6.9. For a connected n-vertex graph G, the height of (X , ≺) is in
O(n3 ).
Proof. Let X be a connected dominating set of a connected graph G. If
G[X] is not an induced path, every vertex in XPof degree at most 2 in G[X]
is contained in at most one half-path. Hence, x∈X wX (x) ≤ |X|. If G[X]
is
Pan induced path, every vertex appears in exactly two half-paths, implying
x∈X wX (x) ≤ 2|X|. Thus
!2
X
X
2
w(X) =
(wX (x)) ≤
wX (x)
≤ 4|X|2 ,
x∈X
x∈X
6.2 A polynomial-time algorithm to find a special connected
dominating set in Pk -free graphs
79
and so the weight of a connected dominating set is in O(n2 ). Since there
are at most n different possible sizes of connected dominating sets of G, the
height of (X , ≺) is in O(n3 ).
Proof of Theorem 6.8. Assume we are given a connected graph G with n
vertices and m edges as input and let k ∈ N be the smallest integer such that
G is Pk -free. Our algorithm works as follows, starting with the connected
dominating set Y := V (G). Its output is a connected dominating set X with
the properties stated in Theorem 6.8.
1. Compute a minimal connected dominating set X ⊆ Y .
2. If G[X] is an induced path, return X and terminate the algorithm.
3. Compute the set NC (X) and the weight wX (x) for every x ∈ NC (X).
4. Order the vertices of NC (X) in non-increasing weight wX , breaking
ties arbitrarily. Let that order be v1 , v2 , . . . , v|NC (X)| .
5. For i from 1 to |NC (X)| do as follows:
(a) Compute a private neighbor yi of vi w.r.t. X.
(b) For j from i + 1 to |NC (X)| do as follows:
i. Check whether Yij := (X ∪ {yi }) \ {vj } is a connected dominating set.
ii. If yes, let Y ← Yij and go to Step 1.
6. Return X and terminate the algorithm.
We remark that the computation of yi in Step 5a is always possible, since xi
is non-cutting in G[X] and X is a minimal connected dominating set.
See Figure 6.2 for an illustration of Step 5(b)ii.
0
2
0
0
0
0
0
0
0
0
0
0
4
0
1
1
Figure 6.2: Before (left) and after (right) an application of Step 5(b)ii. In
the next iteration, the algorithm terminates with the connected dominating
set on the right as output.
The proof is completed by the following sequence of claims.
80
Chapter 6. A characterization of Pk -free graphs
Claim 6.10. When the algorithm terminates, the output X is a connected
dominating set and G[X] is Pk−2 -free or G[X] ∼
= Pk−2 .
Proof. Since Step 1 is applied before the algorithm terminates, X is always a minimal connected dominating set. Since G is Pk -free and hence
(Pk+1 , Ck+1 )-free, G[X] is Pk−1 -free, by Lemma 6.5. If the algorithm terminates with Step 2, it holds that G[X] is isomorphic to P` for some ` 6 k − 2.
Hence, either G[X] ∼
= Pk−2 or G[X] is Pk−2 -free.
Now assume that the algorithm terminates in Step 6. In particular, G[X]
is not an induced path. Suppose for a contradiction that G[X] contains
an induced Pk−2 , say on the vertices x1 , x2 , . . . , xk−2 . Like in the proof of
Lemma 6.6, neither x1 , nor xk−2 can be cutvertices of G[X]. Thus, x1 , xk−2 ∈
NC (X ).
After Step 4, the vertices of NC (X) are ordered v1 , v2 , . . . , v|NC (X)| with
non-increasing weight. Without loss of generality x1 = vi , xk−2 = vj , and
i < j and let yi be the private neighbor of vi that is computed by the
algorithm. As X is returned, the set Yij := (X ∪ {yi }) \ {vj } is in particular
not a connected dominating set, in contradiction to Lemma 6.6. This proves
our claim.
Claim 6.11. Let X be a minimal connected dominating set considered in
some iteration of the algorithm. Assume that the ‘go to’ is called in Step 5(b)ii
because Yij := (X ∪ {yi }) \ {vj } is a connected dominating set. Let X 0 be the
minimal connected dominating set computed in the subsequent Step 1. Then
X ≺ X 0.
Proof. Note that |X| > 2 since the ‘go to’ is called in Step 5(b)ii and G[X]
is not an induced path. So we can assume that every vertex in G[X] has
degree at least 1. Clearly |X 0 | ≤ |X|. If |X 0 | < |X|, X ≺ X 0 by definition.
So we may assume that |X 0 | = |X|, and hence X 0 = Yij . It remains to show
that w(X) < w(X 0 ).
Let z ∈ X \ {vi , vj }. If z has degree at least 2, then wX (z) = 0 and
hence wX 0 (z) > wX (z). Now suppose z has degree 1. Let zx1 x2 · · · xs be a
half-path starting in z. As G[X] is not a path, xs is a cutvertex of G[X]. In
particular, xs 6= vj . Hence, in G[X 0 ], zx1 x2 · · · xs is the initial segment of a
half-path starting in z. In particular, wX 0 (z) ≥ wX (z).
If vi is not a degree-1 vertex of G[X], wX 0 (vi ) = wX (vi ) = 0, and yi vi
is the initial segment of a half-path starting in yi . Hence, wX 0 (yi ) ≥ 1, and
thus
wX 0 (vi ) = 0 and wX 0 (yi ) ≥ wX (vi ) + 1.
(6.1)
If the degree of vi in G[X] is 1, let vi x1 x2 · · · xs be a half-path starting in
vi . Again, xs is a cutvertex of G[X], and so xs 6= vj . Hence, in G[X 0 ],
yi vi x1 x2 · · · xs is the initial segment of a half-path starting in yi . Again (6.1)
holds.
6.2 A polynomial-time algorithm to find a special connected
dominating set in Pk -free graphs
81
Summing up, we see that (6.1) holds, and
wX 0 (z) ≥ wX (z) for every vertex z ∈ X 0 \ {yi , vi }.
(6.2)
We now turn to the vertex vj . First assume that the degree of vj in G[X]
is at least 2, and thus wX (vj ) = 0. Combining this with (6.1) and (6.2)
yields
w(X 0 ) − w(X) ≥ wX 0 (yi )2 − wX (vj )2 > 0,
and so w(X 0 ) − w(X) > 0.
Now assume that vj is a vertex of degree 1 in G[X], and so wX (vj ) ≥ 1.
Let NG[X] (vj ) = {x}. As G[X] is not a path, dG[X] (x) ≥ 2, and so wX (x) =
0. Thus wX 0 (x) = wX (vj ) − 1. Recall that (6.1) and (6.2) hold. We obtain
the following inequality.
w(X 0 ) − w(X) ≥ wX 0 (yi )2 + wX 0 (x)2 − wX (vi )2 − wX (vj )2
= (wX 0 (yi )2 − wX (vi )2 ) − (wX (vj )2 − wX 0 (x)2 )
≥ [(wX (vi ) + 1)2 − wX (vi )2 ] − [wX (vj )2 − (wX (vj ) − 1)2 ]
But wX (vi ) ≥ wX (vj ) implies
(wX (vi ) + 1)2 − wX (vi )2 > wX (vj )2 − (wX (vj ) − 1)2 ,
and thus w(X 0 ) − w(X) > 0 holds as in the previous case.
Hence, X ≺ X 0 , proving our claim.
Claim 6.12. The algorithm terminates in O(n5 (n + m)) time.
Proof. By Claim 6.11, each call of the ‘go to’-step and the subsequent application of Step 1 result in a connected dominating set that is properly larger
in the order ≺. By Lemma 6.9, the height of the poset (X , ≺), and hence
the number of iterations the whole algorithm performs, is in O(n3 ).
The computational complexity of the particular steps remains to be discussed. For this, recall that it can be checked in time O(n + m) whether a
given vertex subset is a connected dominating set. Consequently, Step 1 can
be performed in time O(n(n + m)) by a straightforward greedy procedure.
Step 2 and the computation of the weights in Step 3 can both be performed in linear time using the degree sequence of G[X]. The computation
of the set NC (X) in Step 3 can be done in time O(m) [78].
It remains to discuss the computational complexity of the loop of Step 5.
The computation of a private neighbor in Step 5a can clearly be done in
O(n + m) time. The inner loop of Step 5b performs O(n) checks whether
some vertex set is a connected dominating set, requiring O(n+m) time each.
Hence, Step 5 can be done in O(n2 (n + m)) time.
The overall running time amounts to O(n5 (n + m)), which completes the
proof of both our claim and Theorem 6.8.
82
Chapter 6. A characterization of Pk -free graphs
Chapter 7
2-colorability of hypergraphs
In the previous chapter, we gave a description of the structure of connected
dominating sets in Pk -free graphs. We have shown that any connected Pk free graph admits a connected dominating set whose induced subgraph is
Pk−2 -free or isomorphic to Pk−2 . In fact, any minimum connected dominating set has this property. Loosely speaking, this means that the restricted
structure of connected Pk -free graph results in an even more restricted structure for the induced subgraph of their minimum connected dominating sets.
We now show how to apply these results to a computational problem. A
2-coloring of a hypergraph assigns to each vertex one of two colors, such
that each hyperedge contains vertices of both colors. The Hypergraph
2-Colorability problem is to decide whether a given hypergraph admits
a 2-coloring. In fact, we prove that Hypergraph 2-Colorability, an
N P -complete problem in general, can be solved in polynomial time for hypergraphs whose vertex-hyperedge incidence graph is P7 -free.
7.1
Classical results on hypergraph colorability
Many results are known on the 2-colorability problem in hypergraphs, of
which we present a brief summary. This section is split into two parts: the
former concerns sufficient conditions while the latter is about complexity.
Since our complexity result concerns hypergraphs with conditions on the
vertex-hyperedge incidence graph, we investigate known results with other
conditions on it.
7.1.1
Sufficient conditions
Firstly, conditions are imposed on the size of hyperedges, i.e. on the degree
of vertices corresponding to hyperedges in the incidence graph.
Erdős and Hajnal [56] proved the following theorem.
Theorem 7.1 ( Erdős, Hajnal [56]). All k-uniform hypergraphs with fewer
83
84
Chapter 7. 2-colorability of hypergraphs
than 2k−1 hyperedges is 2-colorable. Moreover, there exists a k-uniform hypergraph with at least k 2 2k−1 hyperedges which is not 2-colorable.
Beck [13, 14] and later Spencer [117], with a shorter proof, showed the
following theorem.
Theorem 7.2 (Beck [13, 14], Spencer [117]). Every k-uniform hypergraph
1
with at most k 3 −O(1) 2k hyperedges is 2-colorable.
Afterward, Radhakrishnan and Srinivasan [103] improved
the previous
q
k
k
upper bound on the number of hyperedges to 0.7
ln(k) 2 . Recently,
Pluhar [102] proved with a new technique a weaker result similar to Theorem 7.2 with the bound ck 1/4 2k .
Moreover, we can also consider hypergraphs with constraints on the degree of all vertices in the incidence graph, especially the d-uniform d-regular
hypergraphs. In these hypergraphs, every hyperedge contains exactly d vertices and each vertex is contained in exactly d hyperedges, i.e. the incidence
graph is d-regular.
In the book called ‘The probabilistic method’ [5], Alon and Spencer
explained the following theorem for hypergraphs with d-regular incidence
graph.
Theorem 7.3 (Alon, Spencer [5]). For d > 9, every d-uniform d-regular
hypergraph is 2-colorable.
Alon and Bregman [3] improved Theorem 7.3 by investigating the case
d = 8. It was implicitly known in literature that Theorem 7.3 was true for
all d > 4, as remarked by Vishwanathan [123] even though we had not seen
it explicitly proved until Henning and Yeo [76] provided a short proof, using
Thomassen’s result [118]. As noticed by Alon and Bregman the result is not
true when d = 3, as may be seen by considering the Fano plane illustrated
by Figure 7.1.
Figure 7.1: The Fano plane.
7.1 Classical results on hypergraph colorability
85
However, Henning and Yeo [76] characterized the 3-uniform 3-regular
hypergraphs in terms of 2-colorability, while Person and Schacht [101] showed
that almost all hypergraphs without Fano planes are 2-colorable.
Theorem 7.4 (Henning, Yeo [76]). Every connected 3-uniform 3-regular
hypergraph is either 2-colorable or becomes 2-colorable after deleting any hyperedge from it.
Finally, Vishwanathan gave another type of constraint on the incidence
graph: the number of quasi-matchings. If G = (A∪B, E) is a bipartite graph
with |A| > |B|, a quasi-matching is a subgraph of G in which the degree of
every vertex in A is one and the degree of every vertex in B is positive. Note
that when |A| = |B|, this is a perfect matching.
Theorem 7.5 (Vishwanathan [123]). Let H be a k-uniform hypergraph.
If the number of quasi-matchings in the incidence graph is different from
0 modulo k then H is 2-colorable.
7.1.2
Computational complexity results
Garey and Johnson [62, p. 221] mentioned that Hypergraph 2-Colorability is N P -complete in general, even for hypergraphs of rank at most 3,
i.e. hypergraphs where the cardinality of any hyperedges is at most 3, as
proved by Lovász [94]. However, it is solvable in polynomial time for hypergraphs of rank 2, i.e. graphs. Moreover, coloring a 2-colorable hypergraph
remains hard as the following result points out.
Theorem 7.6 (Guruswami, Hastad, Sudan [72]). For any constant c, it is
N P -hard to color a 2-colorable 4-uniform hypergraph with c colors.
Latter, Dinur, Regev and Smyth [51] generalized Theorem 7.6 to the class
of 2-colorable 3-uniform hypergraphs.
Theorem 7.7 (Dinur, Regev, Smyth [51]). For any integers k > 3, c2 >
c1 > 1, coloring a k-uniform c1 -colorable hypergraph with c2 colors is N P hard.
Finally, van ’t Hof and Paulusma [122] found a class of hypergraphs where
Hypergraph 2-Colorability problem can be solved in polynomial-time.
Theorem 7.8 (van ’t Hof, Paulusma [122]). Hypergraph 2-Colorability is solvable in polynomial time for hypergraphs with P6 -free incidence
graphs
Firstly, van ’t Hof and Paulusma [122] showed that we can only consider
clutter hypergraph: hypergraph for which no two hyperedges are comparable
with respect to inclusion. Then they applied their characterization of P6 -free
86
Chapter 7. 2-colorability of hypergraphs
graphs on the bipartite incidence graph, i.e. there is a connected dominating
set which is either isomorphic to C6 or contains a complete bipartite subgraph. Finally, they dealt each case in a very short proof. In the further
section, we settle similarly the case of hypergraphs with P7 -free incidence
graphs.
7.2
2-colorability of certain hypergraphs
Our following result is a double application of the characterization of Pk -free
graphs, proved in Chapter 6.
Theorem 7.9. Hypergraph 2-Colorability can be solved in polynomial
time for hypergraphs with P7 -free incidence graphs. If it exists, a 2-coloring
can be computed in polynomial time.
First of all, we simplify the structure of the hypergraph H = (V, E)
by considering only clutter, as proved by van ’t Hof and Paulusma [122].
Secondly, we apply our characterization of Pk -free graphs on the incidence
graph to obtain a connected P5 -free dominating set X, since odd cycle does
not exist in bipartite graph. Contrary to the argumentation of van ’t Hof
and Paulusma, we need to apply again our characterization on G[X] and we
bring out a dominating edge ve in the bipartite graph G[X], where v ∈ V
and e ∈ E. Finally, we settle the proof by dealing cases depending on the
cardinality of e.
Proof of Theorem 7.9. Let H = (V, E) be a hypergraph whose incidence
graph is P7 -free. We denote a 2-coloring of H by (A, B), where A, B ⊆ V
are two non-empty sets with A ∪ B = V , each of which intersects every
hyperedge.
The following observation was proven by van ’t Hof and Paulusma [122].
In order to be self-contained, we give a quick proof of it.
Claim 7.10. We may assume that H is a clutter, i.e. no hyperedge is included in another.
Proof. Assume there are hyperedges e, f ∈ E with e ⊆ f . We can detect
such a pair of hyperedges in polynomial time.
Every 2-coloring of H is a 2-coloring of the hypergraph H 0 = (V, E \ {f })
in particular. If (A, B) is a 2-coloring of H 0 , it holds that e ∩ A 6= ∅ and
e ∩ B 6= ∅. Thus, f ∩ A 6= ∅ and f ∩ B 6= ∅, and so (A, B) is a 2-coloring of
H.
So we may delete, for every such pair e, f ∈ E with e ⊆ f the hyperedge
f from H. It is clear that the resulting hypergraph is a clutter, and its
incidence graph is still P7 -free. This proves Claim 7.10.
7.2 2-colorability of certain hypergraphs
87
Although immediate, Claim 7.10 considerably simplifies the argumentation of the following proof. We now assume that H is a clutter. Moreover, we
may assume that H is connected, which is, its incidence graph is connected.
In the following, we prove a sequence of claims that discuss all relevant cases
for the 2-coloring problem. We give the polynomial-time algorithm along
the way.
Let G be the incidence graph of H. Since we are searching for a 2-coloring,
we may assume that |NG (f )| ≥ 2 for every f ∈ E. By Theorem 6.7, there is
a connected dominating set X of G such that G[X] is P5 -free or G[X] ∼
= C7 .
However, the latter case contradicts the fact that G is bipartite. So, G[X] is
a connected P5 -free graph.
Using Theorem 6.7 again, we see that G[X] has a connected dominating
set Y such that G[Y ] is P3 -free. That is, there is a pair of adjacent vertices,
say v ∈ V and e ∈ E, that together dominate G[X]. In particular, e is
adjacent to vertices of X from the ground set V in the bipartite graph G[X],
which dominate the hyperedge set in G, i.e. e intersects every other hyperedge
of H. It is clear that we can compute such a hyperedge in polynomial time.
Claim 7.11. If there is a proper subset Y ⊂ e which dominates E in G,
(Y, V \ Y ) is a 2-coloring of H.
Proof. Let f ∈ E be arbitrary. By assumption, f ∩ Y 6= ∅. Since H is a
clutter, f 6⊆ e, and thus f 6⊆ Y . Hence, f \ Y 6= ∅, proving Claim 7.11.
Indeed, it can be checked in polynomial time whether there is a proper
subset Y ⊂ e that dominates E. (If so, Y is found in polynomial time, too.)
In view of Claim 7.11, we may assume that no proper subset of e dominates
E.
We now make a distinction between the cases |e| = 2 and |e| ≥ 3. Let
us first assume that |e| = 2, say e = {x, y}. Since H is a clutter, every
hyperedge f of H contains either x or y. Let Ex , Ey ⊆ E \ {e} such that
every f ∈ Ex contains x, every g ∈ Ey contains y, and Ex ∪ Ey = E \ {e}.
If |Ex | = 0, every hyperedge contains y and, as H is a clutter, another
vertex. Thus a 2-coloring of H is given by ({y}, V \ {y}). By symmetry,
we may now assume that |Ex |, |Ey | ≥ 1. Observe that, if |Ex | = 1, say
Ex = {f }, H is 2-colorable if and only if there is a vertex v ∈ f such that
({v, y}, V \ {v, y}) is a 2-coloring of H. Suppose that H is 2-colorable. If for
every vertex v ∈ f , ({v, y}, V \ {v, y}) is not a 2-coloring of H, there exists
a hyperedge ev = {v, y} for each such vertex v. Let now v be an arbitrary
vertex in f . Since e = {x, y} and ev = {y, v} are two hyperedges of H,
then x and v must have the same color, and so there is a vertex v 0 ∈ f with
the second color. Then, the hyperedge ev0 = {v 0 , y} is monochromatic, a
contradiction. This condition can clearly be checked in polynomial time.
Now let |Ex |, |Ey | ≥ 2. We show next that H admits a 2-coloring. To see
this, pick any f ∈ Ex and g ∈ Ey . Since H is a clutter, f \ e, g \ e 6= ∅. Pick
88
Chapter 7. 2-colorability of hypergraphs
any u ∈ f \ e and v ∈ g \ e. If f v, gu 6∈ E(G), G[{u, f, x, e, y, g, v}] ∼
= P7 ,
a contradiction. As u and v were arbitrary, it must be that f \ e ⊆ g \ e or
g \ e ⊆ f \ e.
Now let f, f 0 ∈ Ex and g ∈ Ey be three mutually distinct hyperedges.
As shown above, the sets f \ e, f 0 \ e are comparable to g \ e. Since H is a
clutter, f \ e is not comparable to f 0 \ e. Hence, either f \ e, f 0 \ e ⊆ g, or
g \ e ⊆ f, f 0 .
S
T In the first case, f \ e ⊆ g for any f ∈ Ex , g ∈ Ey . Thus, ( f ∈Ex f ) \ e ⊆
g. Since H is a clutter, every g ∈ Ey has a neighbor outside the set
g∈EyT
{y} ∪ g∈Ey g. Hence,
({y} ∪
\
g, V \ ({y} ∪
g∈Ey
\
g))
g∈Ey
is a 2-coloring of H. The second case, g \ e ⊆ f, f 0 , is dealt with in a similar
fashion.
So we may assume |e| ≥ 3. Since no proper subset of e dominates E in
G, the following assertion holds: for every x ∈ e there is a hyperedge fx such
that fx ∩ e = {x}.
Claim 7.12. For all x, y ∈ e, fx \ e = fy \ e.
Proof. Let x, y ∈ e. The case where x = y is trivial. So we may assume that
x 6= y.
Suppose that there is a vertex z ∈ fx \ (e ∪ fy ). If there is a vertex
0
z ∈ fy \ (e ∪ fx ), G[{z, fx , x, e, y, fy , z 0 }] ∼
= P7 , a contradiction. Thus,
fy \ (e ∪ fx ) = ∅, and so fy \ e ⊆ fx \ e.
Since H is a clutter, there is a vertex u ∈ fy \ e. As fy \ e ⊆ fx \ e,
u ∈ (fx ∩ fy ) \ e. Since |e| ≥ 3, there is a vertex v ∈ e \ {x, y}. But then
G[{z, fx , u, fy , y, e, v}] ∼
= P7 , a contradiction.
So, fx \ (e ∪ fy ) = ∅ and, for symmetry, fy \ (e ∪ fx ) = ∅. This proves
Claim 7.12. For an illustration, see Figure 7.2.
fy
fx
z
u
x
e
y
v
Figure 7.2: The situation in the proof of Claim 7.12.
Claim 7.13. If |fx \ e| = 1 for some x ∈ e, H does not admit a 2-coloring.
7.2 2-colorability of certain hypergraphs
89
Proof. Assume that |fx \ e| = 1 for some x ∈ e. By Claim 7.12, there is a
vertex v ∈ V such that fy \ e = {v} for all y ∈ e.
Suppose that (A, B) is a 2-coloring of H. We may assume that v ∈ A.
Since for every z ∈ e, fz ∩ B 6= ∅, e ⊆ B holds, a contradiction. So Claim
7.13 holds.
It can be checked in polynomial time whether |fx \ e| = 1 for some x ∈ e.
In view of Claim 7.12 and Claim 7.13, we may now assume that |fx \ e| ≥ 2
for all x ∈ e.
Claim 7.14. Let x, y ∈ e be two arbitrary, distinct vertices and let z ∈ fx \e.
A 2-coloring of H is given by ({x, y, z}, V \ {x, y, z}).
Proof. Let x, y, z be chosen according to the claim. Suppose that ({x, y, z},
V \ {x, y, z}) is not a 2-coloring of H. Thus there is a hyperedge f with
f ⊆ {x, y, z} or f ∩ {x, y, z} = ∅.
Let us first assume f ⊆ {x, y, z}. In particular, |f \ e| ≤ 1. Since
|fx0 \ e| ≥ 2 for all x0 ∈ e, we know that |f ∩ e| =
6 1. As NG (e) dominates
the set E, |f ∩ e| ≥ 2 and so x, y ∈ f . Since H is a clutter, f 6⊆ e, and so
f = {x, y, z}.
By assumption, |fx \ e| ≥ 2, and so there is a vertex z 0 ∈ fx \ (e ∪
{z}). Moreover, since |e| ≥ 3, there is a vertex x0 ∈ E \ {x, y}. But then
G[{z 0 , fx , z, f, y, e, x0 }] ∼
= P7 , a contradiction.
So we may assume f ∩ {x, y, z} = ∅. As NG (e) dominates E, e ∩ f 6= ∅.
Let x0 ∈ e ∩ f . As H is not a clutter, there is some z 0 ∈ f \ e. This situation
is illustrated in Figure 7.3.
f
z0
fx
e
x0
y
x
z
Figure 7.3: The situation in the proof of Claim 7.14. The dashed edge is
optional.
If fx z 0 ∈ E(G), G[{z, fx , z 0 , f, x0 , e, y}] ∼
= P7 , a contradiction. Otherwise,
G[{z, fx , x, e, x0 , f, z 0 }] ∼
= P7 , another contradiction. This proves Claim 7.14.
Clearly, a 2-coloring as provided by Claim 7.14 can be constructed efficiently. This completes the proof.
90
Chapter 7. 2-colorability of hypergraphs
Chapter 8
The Pk -hitting set problem
For a fixed k > 2, a Pk -hitting set of a graph G is a set X of vertices
such that G − X does not contain any path on k vertices as a subgraph.
In other words, X is a Pk -hitting set if and only if X meets the vertex
set of every path on k vertices in G. Naturally, we are interested in the
minimum size of a Pk -hitting set, the Pk -hitting number denoted by ψk (G).
The Pk -hitting set problem is motivated by applications, e.g., to the wireless
sensor networks [66,99,100,124]. In this chapter, we focus on computational
aspects, especially on approximation algorithms.
Clearly, the P2 -hitting set problem is the well-known vertex cover problem. Since the vertex cover problem is N P -complete, it is not surprising that
so is the Pk -hitting set problem, for each k > 3. Brešar, Kardoš, Katrenič and
Semanišin [28] reduced the vertex cover problem to the Pk -hitting set problem by adding a pendent path on b k−1
2 c vertices to each vertex of the graph.
This straightforward reduction shows that every α-approximation algorithm
for the Pk -hitting set problem can be turned into an α-approximation algorithm for the vertex cover problem. Hence, obtaining such an algorithm with
α < 2 for some k > 2 is unlikely for general graphs. However, we remark
that in certain classes of graph, one could approximate the Pk -hitting set
problem within a performance ratio smaller than 2.
The case k = 3. By complementary, the dissociation set of a graph corresponds to a P3 -hitting set. Indeed, as defined by several authors [22, 68,
81, 125], the dissociation set of a graph G is a vertex subset of V (G) whose
induced subgraph has maximum degree 1. Thus, the dissociation number ,
which is the maximum size of a dissociation set, is the difference between
the number of vertices and ψ3 (G).
As observed by Tu and Zhou [120,121], the primal-dual 2-approximation
algorithm [12, 15, 46] for the feedback vertex set problem readily gives a 2approximation for the P3 -hitting set problem when adapted to this case. At
a high level point of view, this is essentially due to the fact that graphs
91
92
Chapter 8. The Pk -hitting set problem
not containing P3 as a subgraph are very restricted forests (every connected
component being isomorphic to K1 or K2 ). Tu and Yang [119] investigated
the P3 -hitting set problem for cubic graphs. For this special class of graphs,
they proved the N P -completeness of the P3 -hitting set problem, gave sharp
lower and upper bounds on ψ3 (G) and proposed a 1.57-approximation algorithm. Kardoš, Katrenič and Schiermeyer [81] presented an exact algorithm
with a running time O(1.5171n ) for the P3 -hitting set problem.
The case k = 4. In this chapter, we explain a 3-approximation algorithm
for the P4 -hitting set problem. Our algorithm follows a primal-dual approach
and is very much inspired by the 2-approximation for the feedback vertex set
problem [12, 15, 46]. We note that, as is usual with primal-dual algorithms,
our 3-approximation algorithm also works for the vertex-weighted version of
the problem.
The general case. Brešar et al. [28] described a linear-time exact algorithm in the class of trees and proved the upper bound ψk (T ) 6 |V (T )|/k,
for any tree T . For an arbitrary graph G, they also found various upper
and lower bounds on ψk (G) depending on the number of edges, the degree
of vertices or other invariants and even exact value in certain classes of
graphs [27, 28].
For k > 4, one difficulty of the Pk -hitting set problem is that graphs containing no Pk as a subgraph may have cycles. Note that a k-approximation
can trivially be obtained by taking all vertices in an inclusion-wise maximal packing of vertex-disjoint subgraphs each isomorphic to Pk . However,
nothing better than a k-approximation is known for the problem in general
graphs when k > 4, but for a variant of the feedback vertex set problem,
Guruswami and Lee [73] proved recently strong N P -hardness of approximation result: under the Unique Games Conjecture, for any integer k > 3, and
ε > 0, it is hard to find a (k − ε)-approximate solution to the problem of
intersecting every cycle of length at most k.
A natural question arises: is there an ε > 0 such that the Pk -hitting set
problem can be approximated to within a performance ratio of (1 − ε)k in
polynomial time for every fixed k > 4?
8.1
P4 -hitting set for general graphs
The main result of this chapter is the following theorem.
Theorem 8.1. The P4 -hitting set problem admits a 3-approximation algorithm.
Our algorithm is based on the primal-dual method, see [65] for a survey
of that method. The starting point is the following linear programming (LP)
8.1 P4 -hitting set for general graphs
93
relaxation of the problem. Below, P4 (G) denotes the set of subgraphs P ⊆ G
that are isomorphic to P4 , E(X) denotes the set of edges in G[X] for every
X ⊂ V (G) and dX (v) denotes the degree of the vertex v in G[X].
X
xv
MIN
v∈V (G)
s.t.
X
∀P ∈ P4 (G);
xv > 1
(1)
v∈V (P )
X
(dS (v) − 1)xv > |E(S)| − |S|
∀∅ ( S ⊆ V (G); (2)
v∈S
∀v ∈ V (G).
xv > 0
The dual of this LP relaxation is:
MAX
X
X
zP
P ∈P4 (G)
S⊆V (G)
s.t.
X
(|E(S)| − |S|) yS +
X
(dS (v) − 1) yS +
S:v∈S
zP 6 1
∀v ∈ V (G);
(3)
P :v∈V (P )
∀S ⊆ V (G);
∀P ∈ P4 (G).
yS > 0
zP > 0
The variables yS (∅ ( S ⊆ V (G)) of the dual correspond to inequalities
(2), which we call sparsity inequalities, while the variables zP (P ∈ P4 (G))
correspond to the covering constraints (1). Before explaining the algorithm,
we verify that the LP given above is indeed a relaxation of our problem.
Lemma 8.2. Let X be a P4 -hitting set of the graph G = (V, E). For any
S ⊆V,
X
(dS (v) − 1) > |E(S)| − |S|.
(8.1)
v∈X∩S
Proof. To see that (8.1) holds, first consider the case S = V (G) = V . If
E(X, V \ X) denotes edges between X and V \ X, we have
X
X
(dV (v) − 1) =
dV (v) − |X|
v∈X
v∈X
= 2|E(X)| + |E(X, V \ X)| − |X|
= |E(G)| + |E(X)| − |E(V \ X)| − |X|
= |E(G)| − |V | + |E(X)| − |E(V \ X)| + |V \ X|
> |E(G)| − |V | − |E(V \ X)| + |V \ X|
> |E(G)| − |V |.
94
Chapter 8. The Pk -hitting set problem
The last inequality is the key inequality here. It stems from the fact that
|E(H)| 6 |V (H)| for every graph H not containing a subgraph isomorphic
to P4 , as is easily checked.
Finally, the validity in the case of an arbitrary subset S of V (G) follows
from the observation that if X is a P4 -hitting set of G then X ∩ S is a
P4 -hitting set of G[S].
Our algorithm essentially follows the standard template from the primaldual method. The main difference compared to, e.g., Chudak, Goemans,
Hochbaum and Williamson [46] is that we have two types of inequalities
(the sparsity constraints and the covering constraints) rather than one.
The algorithm maintains an initially infeasible 0/1-solution x to the primal LP and a feasible solution (y, z) to the dual LP. In each iteration, a
certain dual variable is raised until a dual constraint (3) becomes tight. The
corresponding vertex is then “added” to the primal solution. These steps
are repeated until the primal solution becomes feasible. Finally, a “reversedelete” operation is performed and ensures a certain form of minimality for
the P4 -hitting set X output by the algorithm, which is needed in the analysis.
A subtlety of the algorithm is that we do not allow to increase a dual
variable of the type zP for every P ∈ P4 (G). This is allowed only for
“good” paths P ∈ P4 (G), where P is said to be good in G if for some vertex
v ∈ V (P ), every subgraph of G isomorphic to P4 that includes v also includes
another vertex from P . Observe also that every inclusion-wise minimal P4 hitting set of G avoids the vertex v, and thus contains at most 3 vertices
from P . Also, the non-existence of good paths simplifies the analysis of our
algorithm and imposes a structure on the graph, as explained in the following
lemma. A vertex v ∈ V (G) is irrelevant for G if for every P ∈ P4 (G), v ∈
/
V (P ).
Lemma 8.3. Let G be a graph without irrelevant vertices. If none of the
paths P ∈ P4 (G) is good, then G has minimum degree at least 3.
Proof. Observe that for every vertex v ∈ V (G), there exists a path Pv ∈
P4 (G) such that v ∈ V (Pv ).
Firstly, we suppose that there exists a degree-1 vertex v ∈ V (G). Let Pv
be a corresponding 4-vertex path. Then Pv is a good path in G since every
path P ∈ P4 (G) with v ∈ V (P ) must contain the neighbor of v.
Secondly, assume that G has a degree-2 vertex v. Let u1 , u2 be the
neighbors of v and Pv a corresponding 4-vertex path of v. If {u1 , u2 } * V (Pv )
then Pv contains either u1 or u2 , say u1 . Hence Pv = vu1 xy for some
vertices x, y ∈ V (G). Thus, P = u2 vu1 x is a good path in G, since every
path P ∈ P4 (G) with v ∈ V (P ) must contain either u1 or u2 . Otherwise
{u1 , u2 } ⊆ V (Pv ). Similarly, Pv is a good path in G.
In both cases, we obtain a contradiction. Thus the minimum degree of
G is at least 3.
8.1 P4 -hitting set for general graphs
95
Before proving Theorem 8.1, we prove the helpful following lemma.
Lemma 8.4. Let G be a graph without irrelevant vertices and without good
paths P ∈ P4 (G). For any inclusion-wise minimal P4 -hitting set X of G,
X
(dG (v) − 1) 6 3(|E(G)| − |V (G)|)
(8.2)
v∈X
Proof. By Lemma 8.3, the minimum degree of G is at least 3. Since X is
a P4 -hitting set, G − X is the disjoint union of triangles and stars, where a
vertex and an edge are particular stars. Suppose that there are k ? stars and
k 4 triangles in G − X. Note by ξ(X) the number of edges between X and
V (G) \ X, i.e. ξ(X) = |E(X, V (G) \ X)|. Because a triangle has 3 edges and
a forest on n vertices with s connected components has n − s edges, we have
X
dG (v) = 2|E(G)| − ξ(X) − 2(3k 4 + (|V (G)| − |X| − 3k 4 − k ? )).
v∈X
Hence, (8.2) is equivalent to
|X| + 2k ? + |V (G)| − |E(G)| 6 ξ(X).
(8.3)
Since the minimum degree of G is at least 3,
X
X
2|E(G)| =
dG (v) >
3 = 3|V (G)|.
v∈V (G)
v∈V (G)
Then the left member of (8.3) is upper bounded by:
|X| + 2k ? + |V (G)| − |E(G)| 6 |X| + 2k ? −
We distinguish two cases. Assume that |X| 6
|X| + 2k ? −
|V (G)|
2
|V (G)|
.
2
+ k ? . Thus
|V (G)|
|V (G)|
|V (G)|
6
+ 3k ? −
= 3k ? .
2
2
2
Besides, 3k ? 6 ξ(X) because each star in V (G) \ X sends 3 edges to X since
the minimum degree of G is at least 3.
Now, suppose that |X| > |V (G)|
+ k ? . Since each star contains at least
2
+ 2k ? ,
one vertex, |X| + k ? 6 |V (G)|. Hence |V (G)| > |X| + k ? > |V (G)|
2
|V (G)|
which implies 4 > k ? . Thus,
|X| + 2k ? −
|V (G)|
|V (G)| |V (G)|
6 |X| + 2
−
= |X|
2
4
2
Furthermore, |X| 6 ξ(X) since, by minimality of X, each vertex x in X has a
private path, which is a path Px on 4 vertices such that V (Px )∩X = {x}.
Chapter 8. The Pk -hitting set problem
96
Algorithm 1 Primal-dual 3-approximation algorithm for P4 -hitting set
Require: A graph G
Ensure: A P4 -hitting set X of G
X ← ∅, G0 ← G, ` ← 0
yS ← 0 for every subset S ⊆ V (G)
zP ← 0 for every subgraph P of G isomorphic to P4
while G0 contains a subgraph isomorphic to P4 do
`←`+1
V 0 ← V (G0 )
Remove from G0 every vertex v that is irrelevant for G0
if G0 contains a good P4 in G0 , say P , then
Increase zP until ∃v` ∈ V 0 :
X
X
(dS (v` ) − 1) yS +
zP = 1
S:v` ∈S
else
Increase yV 0 until ∃v` ∈ V 0 :
X
(dS (v` ) − 1) yS +
S:v` ∈S
end if
X ← X ∪ {v` }
G0 ← G0 − v `
end while
for j ← ` downto 1 do
if X − {vj } is a P4 -hitting set then
X ← X − {vj }
end if
end for
return X
P :v` ∈V (P )
X
P :v` ∈V (P )
zP = 1
8.1 P4 -hitting set for general graphs
97
Proof of Theorem 8.1. We show that Algorithm 1 is a 3-approximation. Let
X be the P4 -hitting set output by Algorithm 1.
Firstly, since the “reverse-delete” step deletes redundant vertices in the reverse of the order in which they were added, X ∩V (G0 ) must be an inclusionwise minimal P4 -hitting set of every current graph G0 .
Secondly, since every vertex added in X is tight for the sparsity constraint, we have
X
|X| =
1
v∈X 

X X
X

=
zP 
(dS (v) − 1) yS +
v∈X
=
(?)
6
S:v∈S
P
!:v∈V (P )
X
X
∅(S⊆V (G)
v∈X∩S
X
(dS (v) − 1)
X
=
3
6
3 ψ4 (G)
|P ∩ X| zP
P ∈P4 (G)
X
3(|E(S)| − |S|) yS +
∅(S⊆V
 (G)
X
yS +
3 zP
P ∈P4 (G)

(|E(S)| − |S|) yS +
∅(S⊆V (G)
X
zP 
P ∈P4 (G)
We explain the key inequality (?). For the first sum, we can only consider terms when yS 6= 0, i.e. when S = V (G0 ) for one current graph G0
without irrelevant vertices and without good paths in P4 (G0 ). Accordingly,
Lemma 8.4 implies directly what we want because X ∩ S is an inclusionwise minimal P4 -hitting set of G0 . For the second sum, we only consider
terms when zP 6= 0, i.e. when P is a good path of one current G0 . By definition, this path must have at most 3 vertices in every minimal inclusion0
wise
X ∩ V (G0 ). Finally, we conclude because
P P4 -hitting set of G , e.g. P
∅(S⊆V (G) (|E(S)| − |S|) yS +
P ∈P4 (G) zP is a lower bound on ψ4 (G) by
duality.
Note that our algorithm runs in polynomial time because testing if a
path is good can be done in polynomial time.
98
Chapter 8. The Pk -hitting set problem
Chapter 9
Conclusion and further
research
In this research, several problems were solved, but our reflection has led to
new questions, which concern the algorithmic and structural points of view.
• The price of connectivity
Given a graph G and a constant r, we proved that the problem of
deciding whether the price of connectivity of G is at most r is Θp2 complete, for the vertex cover problem and the dominating set problem.
Then, for both problems, we considered the following distinct decision
problem: for any constant r, given a graph G, determine whether the
price of connectivity for all induced subgraphs of G is at most r. We
characterized the class of graphs with a ‘yes’-answer to the previous
question, which are called PoC-near-perfect graphs, for small values of
r, namely r ∈ [1, 3/2] for both problems. Hence, a natural question is to
know, given a rational number r ∈ (3/2, 2) for the vertex cover problem
and r ∈ (3/2, 3) for the dominating set problem, whether the list of
minimal forbidden graphs is finite for the characterization of PoC-nearperfect graphs with threshold r? In particular, for the dominating set
problem, are C9 , P9 and the graph H from Conjecture 5.9 the only
minimal forbidden graphs for the characterization of PoC-near-perfect
graphs with threshold 2?
For the vertex cover problem, in the class of chordal graphs, we proved
that the class of critical graphs is the same as the class of stronglycritical graphs, which can be characterized by special trees. A natural question arises: for both problems, is it possible to characterize
strongly-critical graphs, and more generally critical graphs? It might
also be interesting to investigate the computational complexity of the
critical-graph recognition.
For the dominating set problem, we noticed that even in the class
99
100
Chapter 9. Conclusion and further research
of trees, critical graphs are not completely known. The computation
of the domination number and the connected domination number is
linear in the class of trees. Hence, the price of connectivity of every
tree can be computed in linear time. However, recognizing whether a
tree is critical is still of unknown computational complexity. From a
structural point of view, is it possible to characterize critical trees?
In this thesis, for the dominating set problem, we studied the price
of connectivity of graphs with diameter 2 and some other constraints.
According to the obtained results, we conjecture that the price of connectivity of graphs with diameter 2 is always bounded by 3/2.
• The characterization of Pk -free graphs and applications
We gave a characterization of Pk -free graphs: a graph G is Pk -free if
and only if every connected induced subgraph of G admits a connected
dominating set which is either Pk−2 -free or isomorphic to Ck . As an application of this characterization, we gave a polynomial-time algorithm
solving the Hypergraph 2-Colorability problem for hypergraphs
with P7 -free incidence graphs. Consequently, a natural question would
be to know whether there is any k for which the Hypergraph 2Colorability problem for hypergraphs with Pk -free incidence graphs
is N P -complete.
• The Pk -hitting set problem for k > 4
We described a 3-approximation algorithm for the P4 -hitting set problem, based on the primal-dual method. Is it possible to push our
algorithm further, for instance by adding other constraints, in order
to obtain a 2-approximation? Is there an ε > 0 such that the Pk hitting set problem can be approximated to within a performance ratio
of (1 − ε)k in polynomial time for every fixed k > 4?
A variant of the Pk -hitting set could be a set of vertices meeting all induced k-vertex paths (called Pk -induced-hitting set). Of course, every
Pk -hitting set is a Pk -induced-hitting set but the converse is not true.
This problem is also N P -complete, since the reduction of Brešar, Kardoš, Katrenič and Semanišin [28] from the vertex cover problem holds.
It might be interesting to design specialized approximation algorithms
for this problem.
To summarize, this thesis is a path from the price of connectivity to the
Pk -hitting set problem, through induced subgraphs.
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Index
H-free, 14
Pk -hitting number, 91
Pk -hitting set, 91
H -hitting number, 15
connected, 15
H -hitting set, 15
connected, 15
minimal, 15
minimum, 15
minimum connected, 15
Θp2 -complete, 11
α-approximable, 11
α-approximation, 11
d-regular
graph, 12
hypergraph, 14
k-partite, 13
k-uniform
hypergraph, 14
2-coloring hypergraph, 83
hypergraph, 14
competitive
algorithm, 19
ratio, 19
complement, 13
complete graph, 13
connected
component, 13
graph, 13
hypergraph, 14
corona, 27
critical graph
domination, 61
vertex cover, 44
cubic, 12
cutset, 13
cutvertex, 13
cycle, 13
acyclic, 13
approximation algorithm, 11
decision problem, 9
degree
graph, 12
hypergraph, 14
maximum, 12
minimum, 12
diameter, 12
disconnected, 13
disjoint union, 13
dissociation
number, 91
set, 91
distance, 12
dominating set
bipartite graph, 14
complete, 14
butterfly, 24
center, 12
central vertex, 12
chordal graph, 14
claw, 14
clique, 13
clutter, 85
cograph, 14
coloring
111
112
paired-, 27
total , 27
domination number
paired-, 27
total, 27
upper, 26
upper paired-, 27
upper total, 27
flower, 40
forest, 13
linear, 13
graph, 12
ground set, 14
heuristic, 11
hypergraph, 14
independence number, 13
independent set, 13
induced
cycle, 13
path, 12
subgraph, 13
internal vertex, 13
invariant, 13
isolated vertex, 12
isomorphic, 13
length
cycle, 13
path, 12
lollipop, 25
eaten simple lollipop, 26
simple lollipop, 26
matching, 14
perfect, 14
quasi-, 85
maximization problem, 9
minimization problem, 9
monochromatic, 14
Moore graph, 14
neighbor, 12
INDEX
neighborhood
closed, 12
open, 12
non-trivial graph, 29
NP, 10
NP-complete, 10
NP-hard, 10
objective value, 9
on-line algorithm, 19
optimization problem, 9
oracle, 10
machine, 10
order, 12
P, 10
parameter, 13
path, 12
pendent vertex, 12
perfect graph, 29
performance ratio, 11
planar graph, 14
PoC-near-perfect graph
domination, 54
vertex cover, 37
PoC-perfect graph
domination, 54
vertex cover, 36
price of
anarchy, 20
connectivity, 19
domination, 22, 49
vertex cover, 20, 31
stability, 20
private
neighbor, 12
path, 95
radius, 12
rank
hypergraph, 14
reduction
polynomial-time, 10
set cover, 9
INDEX
problem, 9
split graph, 14
star, 14
strongly critical graph
domination, 61
vertex cover, 44
subgraph, 12
induced, 12
subtree, 13
tree, 13
113
maximum leaf spanning tree,
13
peculiar tree, 62
spanning tree, 13
special tree
dominating set, 62
vertex cover, 44
trivially perfect graph, 14
vertex-hyperedge incidence graph
hypergraph, 14

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