ON SOME GRAPH MODIFICATION
PROBLEMS RELATED TO GRAPH
PARAMETERS
BERNARD RIES
LAMSADE, Université Paris-Dauphine
Séminaire Optimisation Combinatoire
B. Ries ([email protected])
LAMSADE 2014
1 / 56
CO-AUTHORS
Cristina Bazgan
Université Paris-Dauphine, France
Cédric Bentz
Conservatoire National des Arts et Métiers, France
Marie-Christine Costa
Ecole Nationale Supérieure de Techniques Avancées, France
Dominique de Werra
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Christophe Picouleau
Conservatoire National des Arts et Métiers, France
Daniel Paulusma
Durham University, United Kingdom
Oznur Yasar
Kadir Has University, Turkey
B. Ries ([email protected])
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1
INTRODUCTION
2
EDGE CONTRACTIONS
3
EDGE DELETION
4
EDGE ADDITION
5
RELATED PROBLEMS
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LAMSADE 2014
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GENERAL FORMULATION
A graph modification problem is usually defined as follows:
B. Ries ([email protected])
LAMSADE 2014
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GENERAL FORMULATION
A graph modification problem is usually defined as follows:
We fix a graph class G and a set S of one or more graph operations.
B. Ries ([email protected])
LAMSADE 2014
4 / 56
GENERAL FORMULATION
A graph modification problem is usually defined as follows:
We fix a graph class G and a set S of one or more graph operations.
Input: A graph G = (V , E ) and an integer k ≥ 0.
Question: Can G can be modified into a graph H ∈ G by using at most k
operations from S?
B. Ries ([email protected])
LAMSADE 2014
4 / 56
GENERAL FORMULATION
A graph modification problem is usually defined as follows:
We fix a graph class G and a set S of one or more graph operations.
Input: A graph G = (V , E ) and an integer k ≥ 0.
Question: Can G can be modified into a graph H ∈ G by using at most k
operations from S?
Examples
(i) G = family of bipartite graphs; S = {edge deletion};
B. Ries ([email protected])
LAMSADE 2014
4 / 56
GENERAL FORMULATION
A graph modification problem is usually defined as follows:
We fix a graph class G and a set S of one or more graph operations.
Input: A graph G = (V , E ) and an integer k ≥ 0.
Question: Can G can be modified into a graph H ∈ G by using at most k
operations from S?
Examples
(i) G = family of bipartite graphs; S = {edge deletion};
(ii) G = family of bipartite graphs; S = {contraction};
B. Ries ([email protected])
LAMSADE 2014
4 / 56
GENERAL FORMULATION
A graph modification problem is usually defined as follows:
We fix a graph class G and a set S of one or more graph operations.
Input: A graph G = (V , E ) and an integer k ≥ 0.
Question: Can G can be modified into a graph H ∈ G by using at most k
operations from S?
Examples
(i) G = family of bipartite graphs; S = {edge deletion};
(ii) G = family of bipartite graphs; S = {contraction};
(iii) G = family of cographs; S = {vertex deletion};
B. Ries ([email protected])
LAMSADE 2014
4 / 56
GENERAL FORMULATION
A graph modification problem is usually defined as follows:
We fix a graph class G and a set S of one or more graph operations.
Input: A graph G = (V , E ) and an integer k ≥ 0.
Question: Can G can be modified into a graph H ∈ G by using at most k
operations from S?
Examples
(i) G = family of bipartite graphs; S = {edge deletion};
(ii) G = family of bipartite graphs; S = {contraction};
(iii) G = family of cographs; S = {vertex deletion};
(iv) G = family of chordal graphs; S = {edge addition}.
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GENERAL FORMULATION
Instead of fixing a particular graph class G, one may want to fix a graph
parameter π.
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GENERAL FORMULATION
Instead of fixing a particular graph class G, one may want to fix a graph
parameter π.
(a) Can we modify a graph G into a graph H using at most k operations
from S such that the stability number of H has a certain value?
B. Ries ([email protected])
LAMSADE 2014
5 / 56
GENERAL FORMULATION
Instead of fixing a particular graph class G, one may want to fix a graph
parameter π.
(a) Can we modify a graph G into a graph H using at most k operations
from S such that the stability number of H has a certain value?
(b) Can we modify a graph G into a graph H using at most k operations
from S such that the chromatic number of H is such that
χ(H) ≤ χ(G ) − d , for some integer d ?
B. Ries ([email protected])
LAMSADE 2014
5 / 56
GENERAL FORMULATION
Instead of fixing a particular graph class G, one may want to fix a graph
parameter π.
(a) Can we modify a graph G into a graph H using at most k operations
from S such that the stability number of H has a certain value?
(b) Can we modify a graph G into a graph H using at most k operations
from S such that the chromatic number of H is such that
χ(H) ≤ χ(G ) − d , for some integer d ?
(c) . . . .
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GENERAL FORMULATION
d -OPERATION BLOCKER(π)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H using at most k
OPERATIONS such that π(H) ≤ π(G ) − d ?
B. Ries ([email protected])
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6 / 56
GENERAL FORMULATION
d -OPERATION BLOCKER(π)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H using at most k
OPERATIONS such that π(H) ≤ π(G ) − d ?
OPERATION BLOCKER(π)
Input: A graph G = (V , E ) and two nonnegative integer k, d .
Question: Can G be modified into a graph H using at most k
OPERATIONS such that π(H) ≤ π(G ) − d ?
B. Ries ([email protected])
LAMSADE 2014
6 / 56
1
INTRODUCTION
2
EDGE CONTRACTIONS
3
EDGE DELETION
4
EDGE ADDITION
5
RELATED PROBLEMS
B. Ries ([email protected])
LAMSADE 2014
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DEFINITIONS
Let G = (V , E ) be an undirected graph.
The contraction of an edge uv in G removes the vertices u and v
from G , and replaces them by a new vertex made adjacent to precisely
those vertices that were adjacent to u or v in G .
f
f
a
a
e
u
⇒
v
b
c
B. Ries ([email protected])
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b
d
c
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DEFINITIONS
Let G = (V , E ) be an undirected graph.
The contraction of an edge uv in G removes the vertices u and v
from G , and replaces them by a new vertex made adjacent to precisely
those vertices that were adjacent to u or v in G .
f
f
a
a
e
u
⇒
v
b
c
e
b
d
c
d
We say that G can be k-contracted into a graph H if G can be
modified into H by a sequence of at most k edge contractions.
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PROBLEMS
Let π ∈ {ω, χ, α}, where ω denotes the clique number, χ denotes the
chromatic number and α denotes the stability number.
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PROBLEMS
Let π ∈ {ω, χ, α}, where ω denotes the clique number, χ denotes the
chromatic number and α denotes the stability number.
d -CONTRACTION BLOCKER(π)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be k-contracted into a graph H such that
π(H) ≤ π(G ) − d ?
B. Ries ([email protected])
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PROBLEMS
Let π ∈ {ω, χ, α}, where ω denotes the clique number, χ denotes the
chromatic number and α denotes the stability number.
d -CONTRACTION BLOCKER(π)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be k-contracted into a graph H such that
π(H) ≤ π(G ) − d ?
CONTRACTION BLOCKER(π)
Input: A graph G = (V , E ) and two nonnegative integers k, d .
Question: Can G be k-contracted into a graph H such that
π(H) ≤ π(G ) − d ?
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FIRST RESULTS
1-CONTRACTION BLOCKER(χ) is NP-complete even for graphs of
chromatic number 3.
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FIRST RESULTS
1-CONTRACTION BLOCKER(χ) is NP-complete even for graphs of
chromatic number 3.
Proof:
BIPARTITE CONTRACTION: can a graph be made bipartite by at
most k edge contractions?
B. Ries ([email protected])
LAMSADE 2014
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FIRST RESULTS
1-CONTRACTION BLOCKER(χ) is NP-complete even for graphs of
chromatic number 3.
Proof:
BIPARTITE CONTRACTION: can a graph be made bipartite by at
most k edge contractions?
1-CONTRACTION BLOCKER(χ) and BIPARTITE CONTRACTION
are equivalent for graphs of chromatic number 3;
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FIRST RESULTS
1-CONTRACTION BLOCKER(χ) is NP-complete even for graphs of
chromatic number 3.
Proof:
BIPARTITE CONTRACTION: can a graph be made bipartite by at
most k edge contractions?
1-CONTRACTION BLOCKER(χ) and BIPARTITE CONTRACTION
are equivalent for graphs of chromatic number 3;
BIPARTITE CONTRACTION is NP-complete for graphs of chromatic
number 3 [Heggernes, Lokshtanov, Paul, van’t Hof, 2013].
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FIRST RESULTS
1-CONTRACTION BLOCKER(α) is NP-complete even for complement of
bipartite graphs.
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FIRST RESULTS
1-CONTRACTION BLOCKER(α) is NP-complete even for complement of
bipartite graphs.
Proof:
s-CLUB CONTRACTION: can a graph be k-contracted into a graph
of diameter at most s?
B. Ries ([email protected])
LAMSADE 2014
11 / 56
FIRST RESULTS
1-CONTRACTION BLOCKER(α) is NP-complete even for complement of
bipartite graphs.
Proof:
s-CLUB CONTRACTION: can a graph be k-contracted into a graph
of diameter at most s?
1-CONTRACTION BLOCKER(α) and 1-CLUB CONTRACTION are
equivalent for cobipartite graphs;
B. Ries ([email protected])
LAMSADE 2014
11 / 56
FIRST RESULTS
1-CONTRACTION BLOCKER(α) is NP-complete even for complement of
bipartite graphs.
Proof:
s-CLUB CONTRACTION: can a graph be k-contracted into a graph
of diameter at most s?
1-CONTRACTION BLOCKER(α) and 1-CLUB CONTRACTION are
equivalent for cobipartite graphs;
1-CLUB CONTRACTION is NP-complete even for cobipartite graphs
[Golovach, Heggernes, Paul, van’t Hof, 2014].
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FIRST RESULTS
1-CONTRACTION BLOCKER(ω) is NP-complete even for graphs with
clique number 3.
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FIRST RESULTS
1-CONTRACTION BLOCKER(ω) is NP-complete even for graphs with
clique number 3.
Proof:
use a polynomial reduction from the NP-complete problem
ONE-IN-3-SAT.
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P` -FREE GRAPHS
A graph G = (V , E ) is called P` -free if it does not contain any induced
path on ` vertices.
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P` -FREE GRAPHS
A graph G = (V , E ) is called P` -free if it does not contain any induced
path on ` vertices.
Therorem
Let π ∈ {α, χ, ω). Then CONTRACTION BLOCKER(π) restricted to
P` -free graphs is polynomial-time solvable if ` ≤ 4 and NP-complete if
` ≥ 5.
B. Ries ([email protected])
LAMSADE 2014
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P` -FREE GRAPHS
A graph G = (V , E ) is called P` -free if it does not contain any induced
path on ` vertices.
Therorem
Let π ∈ {α, χ, ω). Then CONTRACTION BLOCKER(π) restricted to
P` -free graphs is polynomial-time solvable if ` ≤ 4 and NP-complete if
` ≥ 5.
Proof:
(i) we prove NP-completeness for split graphs, i.e. (2K2 , C4 , C5 )-free
graphs;
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LAMSADE 2014
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P` -FREE GRAPHS
A graph G = (V , E ) is called P` -free if it does not contain any induced
path on ` vertices.
Therorem
Let π ∈ {α, χ, ω). Then CONTRACTION BLOCKER(π) restricted to
P` -free graphs is polynomial-time solvable if ` ≤ 4 and NP-complete if
` ≥ 5.
Proof:
(i) we prove NP-completeness for split graphs, i.e. (2K2 , C4 , C5 )-free
graphs;
(ii) we prove polynomiality for cographs, i.e. P4 -free graphs.
B. Ries ([email protected])
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P` -FREE GRAPHS
Proof:
(i) we prove NP-completeness for split graphs, i.e. (2K2 , C4 , C5 )-free
graphs;
⇒ we use a reduction from RED BLUE DOMINATING SET;
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P` -FREE GRAPHS
Proof:
(i) we prove NP-completeness for split graphs, i.e. (2K2 , C4 , C5 )-free
graphs;
⇒ we use a reduction from RED BLUE DOMINATING SET;
(ii) we prove polynomiality for cographs, i.e. P4 -free graphs;
⇒ we use dynamic programming on the cotree.
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SPLIT GRAPHS
A split graph is a graph whose vertex set can be partitioned into a clique
and a stable set.
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LAMSADE 2014
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SPLIT GRAPHS
A split graph is a graph whose vertex set can be partitioned into a clique
and a stable set.
Theorem
Let π ∈ {ω, χ, α}. Then d -CONTRACTION BLOCKER(π) is polynomially
solvable for split graphs.
B. Ries ([email protected])
LAMSADE 2014
15 / 56
SPLIT GRAPHS
A split graph is a graph whose vertex set can be partitioned into a clique
and a stable set.
Theorem
Let π ∈ {ω, χ, α}. Then d -CONTRACTION BLOCKER(π) is polynomially
solvable for split graphs.
Proof:
1
if χ(G ) ≤ d ⇒ NO;
B. Ries ([email protected])
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15 / 56
SPLIT GRAPHS
A split graph is a graph whose vertex set can be partitioned into a clique
and a stable set.
Theorem
Let π ∈ {ω, χ, α}. Then d -CONTRACTION BLOCKER(π) is polynomially
solvable for split graphs.
Proof:
1
if χ(G ) ≤ d ⇒ NO;
2
if χ(G ) = d + 1, YES ⇔ k ≥ |V | − 1;
B. Ries ([email protected])
LAMSADE 2014
15 / 56
SPLIT GRAPHS
A split graph is a graph whose vertex set can be partitioned into a clique
and a stable set.
Theorem
Let π ∈ {ω, χ, α}. Then d -CONTRACTION BLOCKER(π) is polynomially
solvable for split graphs.
Proof:
1
if χ(G ) ≤ d ⇒ NO;
2
if χ(G ) = d + 1, YES ⇔ k ≥ |V | − 1;
3
if χ(G ) ≥ d + 2:
I
if k < d ⇒ NO;
B. Ries ([email protected])
LAMSADE 2014
15 / 56
SPLIT GRAPHS
A split graph is a graph whose vertex set can be partitioned into a clique
and a stable set.
Theorem
Let π ∈ {ω, χ, α}. Then d -CONTRACTION BLOCKER(π) is polynomially
solvable for split graphs.
Proof:
1
if χ(G ) ≤ d ⇒ NO;
2
if χ(G ) = d + 1, YES ⇔ k ≥ |V | − 1;
3
if χ(G ) ≥ d + 2:
I
I
if k < d ⇒ NO;
if k = d ⇒ try all possible sequences of at most k edge contractions;
B. Ries ([email protected])
LAMSADE 2014
15 / 56
SPLIT GRAPHS
A split graph is a graph whose vertex set can be partitioned into a clique
and a stable set.
Theorem
Let π ∈ {ω, χ, α}. Then d -CONTRACTION BLOCKER(π) is polynomially
solvable for split graphs.
Proof:
1
if χ(G ) ≤ d ⇒ NO;
2
if χ(G ) = d + 1, YES ⇔ k ≥ |V | − 1;
3
if χ(G ) ≥ d + 2:
I
I
I
if k < d ⇒ NO;
if k = d ⇒ try all possible sequences of at most k edge contractions;
if k > d ⇒ YES.
B. Ries ([email protected])
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INTERVAL GRAPHS
A graph is an interval graph if one can associate an interval on the real
line with each vertex such that two vertices are adjacent if and only if the
corresponding intervals intersect.
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INTERVAL GRAPHS
A graph is an interval graph if one can associate an interval on the real
line with each vertex such that two vertices are adjacent if and only if the
corresponding intervals intersect.
e
b
d
g
f
a
b
a
c
d
e
f
c
g
B. Ries ([email protected])
LAMSADE 2014
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INTERVAL GRAPHS
Theorem
Let π ∈ {ω, χ}. Then CONTRACTION BLOCKER(π) is polynomially
solvable for interval graphs.
B. Ries ([email protected])
LAMSADE 2014
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INTERVAL GRAPHS
Theorem
Let π ∈ {ω, χ}. Then CONTRACTION BLOCKER(π) is polynomially
solvable for interval graphs.
Proof:
(a) if we contract an edge e in a C4 -free graph G , then
B. Ries ([email protected])
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INTERVAL GRAPHS
Theorem
Let π ∈ {ω, χ}. Then CONTRACTION BLOCKER(π) is polynomially
solvable for interval graphs.
Proof:
(a) if we contract an edge e in a C4 -free graph G , then
1
every maximal clique in G containing e will have its size reduced by
exactly one;
B. Ries ([email protected])
LAMSADE 2014
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INTERVAL GRAPHS
Theorem
Let π ∈ {ω, χ}. Then CONTRACTION BLOCKER(π) is polynomially
solvable for interval graphs.
Proof:
(a) if we contract an edge e in a C4 -free graph G , then
1
every maximal clique in G containing e will have its size reduced by
exactly one;
2
every other maximal clique remains the same;
B. Ries ([email protected])
LAMSADE 2014
17 / 56
INTERVAL GRAPHS
Theorem
Let π ∈ {ω, χ}. Then CONTRACTION BLOCKER(π) is polynomially
solvable for interval graphs.
Proof:
(a) if we contract an edge e in a C4 -free graph G , then
1
every maximal clique in G containing e will have its size reduced by
exactly one;
2
every other maximal clique remains the same;
3
we do not create any new maximal clique.
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INTERVAL GRAPHS
(b) one can always choose the two intervals with the rightmost right
endpoints;
⇒
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INTERVAL GRAPHS
(b) one can always choose the two intervals with the rightmost right
endpoints;
⇒
(c) use the facts that interval graphs are closed under edge contraction
and that the number of maximal cliques is polynomial;
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INTERVAL GRAPHS
(b) one can always choose the two intervals with the rightmost right
endpoints;
⇒
(c) use the facts that interval graphs are closed under edge contraction
and that the number of maximal cliques is polynomial;
(d) use repeatedly arguments (a) and (b).
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INTERVAL GRAPHS
What about CONTRACTION BLOCKER(α)?
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INTERVAL GRAPHS
What about CONTRACTION BLOCKER(α)?
What about d -CONTRACTION BLOCKER(α)?
B. Ries ([email protected])
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1
INTRODUCTION
2
EDGE CONTRACTIONS
3
EDGE DELETION
4
EDGE ADDITION
5
RELATED PROBLEMS
B. Ries ([email protected])
LAMSADE 2014
20 / 56
PROBLEMS
d -EDGE DELETION BLOCKER(χ)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H with at most k edge
deletions such that χ(H) ≤ χ(G ) − d ?
B. Ries ([email protected])
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PROBLEMS
d -EDGE DELETION BLOCKER(χ)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H with at most k edge
deletions such that χ(H) ≤ χ(G ) − d ?
EDGE DELETION BLOCKER(χ)
Input: A graph G = (V , E ) and two nonnegative integers k, d .
Question: Can G be modified into a graph H with at most k edge
deletions such that χ(H) ≤ χ(G ) − d ?
B. Ries ([email protected])
LAMSADE 2014
21 / 56
PROBLEMS
d -EDGE DELETION BLOCKER(χ)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H with at most k edge
deletions such that χ(H) ≤ χ(G ) − d ?
EDGE DELETION BLOCKER(χ)
Input: A graph G = (V , E ) and two nonnegative integers k, d .
Question: Can G be modified into a graph H with at most k edge
deletions such that χ(H) ≤ χ(G ) − d ?
Remark: We may assume that χ(G ) ≥ 3. Indeed, if χ(G ) ≤ 2, then we
need to delete all edges in G since G is bipartite.
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that K is maximal). We may assume that |K | ≥ d + 2.
B. Ries ([email protected])
LAMSADE 2014
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that K is maximal). We may assume that |K | ≥ d + 2.
|K | ≥ 2(d + 1)
1
∃ matching M of size d + 1 in G [K ];
B. Ries ([email protected])
LAMSADE 2014
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that K is maximal). We may assume that |K | ≥ d + 2.
|K | ≥ 2(d + 1)
1
∃ matching M of size d + 1 in G [K ];
2
delete the edges of M
B. Ries ([email protected])
LAMSADE 2014
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that K is maximal). We may assume that |K | ≥ d + 2.
|K | ≥ 2(d + 1)
1
∃ matching M of size d + 1 in G [K ];
2
delete the edges of M ⇒ remaining graph is (K − d )-colorable;
B. Ries ([email protected])
LAMSADE 2014
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that K is maximal). We may assume that |K | ≥ d + 2.
|K | ≥ 2(d + 1)
1
∃ matching M of size d + 1 in G [K ];
2
delete the edges of M ⇒ remaining graph is (K − d )-colorable;
3
⇒ M is a solution of size d + 1.
B. Ries ([email protected])
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
B. Ries ([email protected])
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that K is maximal). We may assume that |K | ≥ d + 2.
|K | < 2(d + 1)
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SPLIT GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that K is maximal). We may assume that |K | ≥ d + 2.
|K | < 2(d + 1)
1
since |K | ≥ d + 2, EK is a solution of size at most d (2d + 1).
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SPLIT GRAPHS
What about EDGE DELETION BLOCKER(χ) in split graphs?
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COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
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COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
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COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
µ(G ) ≥ d + 1
B. Ries ([email protected])
LAMSADE 2014
25 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
µ(G ) ≥ d + 1
1
choose d + 1 edges of M and add the set F of all non-edges between
the saturated vertices;
B. Ries ([email protected])
LAMSADE 2014
25 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
µ(G ) ≥ d + 1
1
choose d + 1 edges of M and add the set F of all non-edges between
the saturated vertices;
2
⇒ clique of size 2d + 2 and thus θ(G + F ) ≤ θ(G ) − d ;
B. Ries ([email protected])
LAMSADE 2014
25 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
µ(G ) ≥ d + 1
1
choose d + 1 edges of M and add the set F of all non-edges between
the saturated vertices;
2
⇒ clique of size 2d + 2 and thus θ(G + F ) ≤ θ(G ) − d ;
3
the set F of non-edges in G corresponds to a solution in G of size at
most 2(d+1)
− (d + 1).
2
B. Ries ([email protected])
LAMSADE 2014
25 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
µ(G ) < d + 1
B. Ries ([email protected])
LAMSADE 2014
26 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
µ(G ) < d + 1
1
if there are at least d + 1 non-saturated vertices, then we add all the
non-edges between d + 1 of them;
B. Ries ([email protected])
LAMSADE 2014
26 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
d -EDGE DELETION BLOCKER(χ) is polynomial-time solvable in
complements of bipartite graphs.
Proof:
Consider a maximum matching in G .
µ(G ) < d + 1
1
if there are at least d + 1 non-saturated vertices, then we add all the
non-edges between d + 1 of them;
2
otherwise, G contains at most d + 2µ(G ) ≤ 3d vertices.
B. Ries ([email protected])
LAMSADE 2014
26 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
EDGE DELETION BLOCKER(χ) is NP-hard in complements of bipartite
graphs.
B. Ries ([email protected])
LAMSADE 2014
27 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
EDGE DELETION BLOCKER(χ) is NP-hard in complements of bipartite
graphs.
Proof:
work with the complement, i.e. a bipartite graph;
B. Ries ([email protected])
LAMSADE 2014
27 / 56
COMPLEMENT OF BIPARTITE GRAPHS
Theorem
EDGE DELETION BLOCKER(χ) is NP-hard in complements of bipartite
graphs.
Proof:
work with the complement, i.e. a bipartite graph;
use a reduction from a special case of P3 -PARTITION.
B. Ries ([email protected])
LAMSADE 2014
27 / 56
1
INTRODUCTION
2
EDGE CONTRACTIONS
3
EDGE DELETION
4
EDGE ADDITION
5
RELATED PROBLEMS
B. Ries ([email protected])
LAMSADE 2014
28 / 56
PROBLEMS
d -EDGE ADDITION BLOCKER(α)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H with at most k edge
additions such that α(H) ≤ α(G ) − d ?
B. Ries ([email protected])
LAMSADE 2014
29 / 56
PROBLEMS
d -EDGE ADDITION BLOCKER(α)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H with at most k edge
additions such that α(H) ≤ α(G ) − d ?
EDGE ADDITION BLOCKER(α)
Input: A graph G = (V , E ) and two nonnegative integers k, d .
Question: Can G be modified into a graph H with at most k edge
additions such that α(H) ≤ α(G ) − d ?
B. Ries ([email protected])
LAMSADE 2014
29 / 56
PROBLEMS
d -EDGE ADDITION BLOCKER(α)
Input: A graph G = (V , E ) and a nonnegative integer k.
Question: Can G be modified into a graph H with at most k edge
additions such that α(H) ≤ α(G ) − d ?
EDGE ADDITION BLOCKER(α)
Input: A graph G = (V , E ) and two nonnegative integers k, d .
Question: Can G be modified into a graph H with at most k edge
additions such that α(H) ≤ α(G ) − d ?
Remark: We may assume that α(G ) ≥ 3. Indeed, if α(G ) ≤ 2, then we
need to add all missing edges to G in order to decrease the stability
number.
B. Ries ([email protected])
LAMSADE 2014
29 / 56
SPLIT GRAPHS
Theorem
d -EDGE ADDITION BLOCKER(α) is polynomial-time solvable in split
graphs.
B. Ries ([email protected])
LAMSADE 2014
30 / 56
SPLIT GRAPHS
Theorem
d -EDGE ADDITION BLOCKER(α) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that S is maximal, i.e. α(G ) = |S|). We may assume that |S| ≥ d + 2.
B. Ries ([email protected])
LAMSADE 2014
30 / 56
SPLIT GRAPHS
Theorem
d -EDGE ADDITION BLOCKER(α) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that S is maximal, i.e. α(G ) = |S|). We may assume that |S| ≥ d + 2.
choose d + 2 vertices in S;
1
add all edges between them ⇒ set F of size
B. Ries ([email protected])
(d+1)(d+2)
;
2
LAMSADE 2014
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SPLIT GRAPHS
Theorem
d -EDGE ADDITION BLOCKER(α) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that S is maximal, i.e. α(G ) = |S|). We may assume that |S| ≥ d + 2.
choose d + 2 vertices in S;
(d+1)(d+2)
;
2
1
add all edges between them ⇒ set F of size
2
the new graph H clearly satisfies α(H) ≤ α(G ) − d ;
B. Ries ([email protected])
LAMSADE 2014
30 / 56
SPLIT GRAPHS
Theorem
d -EDGE ADDITION BLOCKER(α) is polynomial-time solvable in split
graphs.
Proof:
Let G = (V , E ) be a split graph with split partition K , S (we may assume
that S is maximal, i.e. α(G ) = |S|). We may assume that |S| ≥ d + 2.
choose d + 2 vertices in S;
(d+1)(d+2)
;
2
1
add all edges between them ⇒ set F of size
2
the new graph H clearly satisfies α(H) ≤ α(G ) − d ;
3
⇒ F is a solution of bounded size.
B. Ries ([email protected])
LAMSADE 2014
30 / 56
SPLIT GRAPHS
What about EDGE ADDITION BLOCKER(α) in split graphs?
B. Ries ([email protected])
LAMSADE 2014
31 / 56
BIPARTITE GRAPHS
Let G = (B, W , E ) be a bipartite graph, where B and W denote the two
sets of the bipartition.
B. Ries ([email protected])
LAMSADE 2014
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BIPARTITE GRAPHS
Let G = (B, W , E ) be a bipartite graph, where B and W denote the two
sets of the bipartition.
• A vertex v is called forced if every maximum stable set contains v .
(⇒ F)
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BIPARTITE GRAPHS
Let G = (B, W , E ) be a bipartite graph, where B and W denote the two
sets of the bipartition.
• A vertex v is called forced if every maximum stable set contains v .
(⇒ F)
• A vertex v is called excluded if no maximum stable set contains v .
(⇒ E)
B. Ries ([email protected])
LAMSADE 2014
32 / 56
BIPARTITE GRAPHS
Let G = (B, W , E ) be a bipartite graph, where B and W denote the two
sets of the bipartition.
• A vertex v is called forced if every maximum stable set contains v .
(⇒ F)
• A vertex v is called excluded if no maximum stable set contains v .
(⇒ E)
• A vertex which is neither forced nor excluded is called free.
B. Ries ([email protected])
LAMSADE 2014
32 / 56
BIPARTITE GRAPHS
Theorem
Let G = (B, W , E ) be a bipartite graph containing only free vertices. Then
there exists a partition V = (V1 , . . . , Vq ) of B ∪ W such that a stable set
S ⊆ B ∪ W is maximum if and only if for any j ∈ {1, . . . , q} either
S ∩ Vj = B ∩ Vj or S ∩ Vj = W ∩ Vj .
B. Ries ([email protected])
LAMSADE 2014
33 / 56
BIPARTITE GRAPHS
Theorem
Let G = (B, W , E ) be a bipartite graph containing only free vertices. Then
there exists a partition V = (V1 , . . . , Vq ) of B ∪ W such that a stable set
S ⊆ B ∪ W is maximum if and only if for any j ∈ {1, . . . , q} either
S ∩ Vj = B ∩ Vj or S ∩ Vj = W ∩ Vj .
We will call a partition V a good partition. For such a partition the
following properties hold:
B. Ries ([email protected])
LAMSADE 2014
33 / 56
BIPARTITE GRAPHS
Theorem
Let G = (B, W , E ) be a bipartite graph containing only free vertices. Then
there exists a partition V = (V1 , . . . , Vq ) of B ∪ W such that a stable set
S ⊆ B ∪ W is maximum if and only if for any j ∈ {1, . . . , q} either
S ∩ Vj = B ∩ Vj or S ∩ Vj = W ∩ Vj .
We will call a partition V a good partition. For such a partition the
following properties hold:
(i) V can be obtained in polynomial time;
B. Ries ([email protected])
LAMSADE 2014
33 / 56
BIPARTITE GRAPHS
Theorem
Let G = (B, W , E ) be a bipartite graph containing only free vertices. Then
there exists a partition V = (V1 , . . . , Vq ) of B ∪ W such that a stable set
S ⊆ B ∪ W is maximum if and only if for any j ∈ {1, . . . , q} either
S ∩ Vj = B ∩ Vj or S ∩ Vj = W ∩ Vj .
We will call a partition V a good partition. For such a partition the
following properties hold:
(i) V can be obtained in polynomial time;
(ii) each graph G [Vi ] is connected and in addition we have
|W ∩ Vi | = |B ∩ Vi |, for i = 1 . . . , p; notice that this implies that the
cardinality of each set Vi is even and at least two;
B. Ries ([email protected])
LAMSADE 2014
33 / 56
BIPARTITE GRAPHS
Theorem
Let G = (B, W , E ) be a bipartite graph containing only free vertices. Then
there exists a partition V = (V1 , . . . , Vq ) of B ∪ W such that a stable set
S ⊆ B ∪ W is maximum if and only if for any j ∈ {1, . . . , q} either
S ∩ Vj = B ∩ Vj or S ∩ Vj = W ∩ Vj .
We will call a partition V a good partition. For such a partition the
following properties hold:
(i) V can be obtained in polynomial time;
(ii) each graph G [Vi ] is connected and in addition we have
|W ∩ Vi | = |B ∩ Vi |, for i = 1 . . . , p; notice that this implies that the
cardinality of each set Vi is even and at least two;
(iii) if there exists an edge between two vertices u, v such that u ∈ Vi ∩ B
and v ∈ Vj ∩ W , i 6= j, then there exists no edge between vertices x, y
such that x ∈ Vi ∩ W and y ∈ Vj ∩ B.
B. Ries ([email protected])
LAMSADE 2014
33 / 56
BIPARTITE GRAPHS
Illustration:
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BIPARTITE GRAPHS
Illustration:
V1
V2
V3
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BIPARTITE GRAPHS
Let b1 (G ) be the minimum number of edges one has to add to G to
make its stability number decrease by 1.
B. Ries ([email protected])
LAMSADE 2014
36 / 56
BIPARTITE GRAPHS
Let b1 (G ) be the minimum number of edges one has to add to G to
make its stability number decrease by 1.
Theorem
1-EDGE ADDITION BLOCKER(α) is polynomial time solvable in bipartite
graphs.
B. Ries ([email protected])
LAMSADE 2014
36 / 56
BIPARTITE GRAPHS
Let b1 (G ) be the minimum number of edges one has to add to G to
make its stability number decrease by 1.
Theorem
1-EDGE ADDITION BLOCKER(α) is polynomial time solvable in bipartite
graphs. If V = {V1 , . . . , Vq } is a good partition of V \ (F ∪ E), then
(a) b1 (G ) = 1 if and only if |F| ≥ 2;
(b) b1 (G ) = 2 if and only if |F| ≤ 1 and
(b1)
(b2)
(b3)
(b4)
either |F| = 1;
or ∃Vi ∈ V such that |Vi | ≥ 4;
or G contains H1 as a subgraph;
or G contains H2 as a subgraph;
(c) in all remaining cases:
(c1) b1 (G ) = 3 if and only if there exist Vi , Vj ∈ V, i 6= j, such that x ∈ Vi ,
y ∈ Vj and xy ∈ E ;
(c2) b1 (G ) = 4 otherwise.
B. Ries ([email protected])
LAMSADE 2014
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BIPARTITE GRAPHS
H1
Vi1
Vi2
Vi3
Vi3
H2
Vi1
Vi2
Vi4
B. Ries ([email protected])
LAMSADE 2014
37 / 56
BIPARTITE GRAPHS
What about d -EDGE ADDITION BLOCKER(α), for d ≥ 2?
B. Ries ([email protected])
LAMSADE 2014
38 / 56
BIPARTITE GRAPHS
What about d -EDGE ADDITION BLOCKER(α), for d ≥ 2?
What about EDGE ADDITION BLOCKER(α)?
B. Ries ([email protected])
LAMSADE 2014
38 / 56
1
INTRODUCTION
2
EDGE CONTRACTIONS
3
EDGE DELETION
4
EDGE ADDITION
5
RELATED PROBLEMS
B. Ries ([email protected])
LAMSADE 2014
39 / 56
BLOCKERS
We just saw the notion of blocker:
B. Ries ([email protected])
LAMSADE 2014
40 / 56
BLOCKERS
We just saw the notion of blocker:
a set of edges we need to contract in order to make the stability
(or chromatic or clique) number decrease;
B. Ries ([email protected])
LAMSADE 2014
40 / 56
BLOCKERS
We just saw the notion of blocker:
a set of edges we need to contract in order to make the stability
(or chromatic or clique) number decrease;
a set of edges we need to delete in order to make the chromatic
number decrease;
B. Ries ([email protected])
LAMSADE 2014
40 / 56
BLOCKERS
We just saw the notion of blocker:
a set of edges we need to contract in order to make the stability
(or chromatic or clique) number decrease;
a set of edges we need to delete in order to make the chromatic
number decrease;
a set of edges we need to add in order to make the stability
number decrease;
B. Ries ([email protected])
LAMSADE 2014
40 / 56
BLOCKERS
We just saw the notion of blocker:
a set of edges we need to contract in order to make the stability
(or chromatic or clique) number decrease;
a set of edges we need to delete in order to make the chromatic
number decrease;
a set of edges we need to add in order to make the stability
number decrease;
a set of vertices we need to delete in order to make the stability
number decrease;
B. Ries ([email protected])
LAMSADE 2014
40 / 56
BLOCKERS
We just saw the notion of blocker:
a set of edges we need to contract in order to make the stability
(or chromatic or clique) number decrease;
a set of edges we need to delete in order to make the chromatic
number decrease;
a set of edges we need to add in order to make the stability
number decrease;
a set of vertices we need to delete in order to make the stability
number decrease;
... .
B. Ries ([email protected])
LAMSADE 2014
40 / 56
BLOCKERS
In case, one has to delete vertices to make the stability number decrease,
one may also ask the following:
B. Ries ([email protected])
LAMSADE 2014
41 / 56
BLOCKERS
In case, one has to delete vertices to make the stability number decrease,
one may also ask the following:
Given a graph G , does there exist a set of at most k vertices such that
every maximum stable set contains at least d of them?
B. Ries ([email protected])
LAMSADE 2014
41 / 56
BLOCKERS
In case, one has to delete vertices to make the stability number decrease,
one may also ask the following:
Given a graph G , does there exist a set of at most k vertices such that
every maximum stable set contains at least d of them?
⇒ notion of d -transversals
B. Ries ([email protected])
LAMSADE 2014
41 / 56
BLOCKERS
In case, one has to delete vertices to make the stability number decrease,
one may also ask the following:
Given a graph G , does there exist a set of at most k vertices such that
every maximum stable set contains at least d of them?
⇒ notion of d -transversals
Are these two notions the same?
B. Ries ([email protected])
LAMSADE 2014
41 / 56
DEFINITIONS
Minimum d-transversal
Let G = (V , E ) be a weighted graph. For every vertex v ∈ V , consider a
cost c(v ) ≥ 0. Let d ≥ 1 be an integer.
B. Ries ([email protected])
LAMSADE 2014
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DEFINITIONS
Minimum d-transversal
Let G = (V , E ) be a weighted graph. For every vertex v ∈ V , consider a
cost c(v ) ≥ 0. Let d ≥ 1 be an integer.
A d-transversal T w.r. to S(G ) is a set T ⊆ V such that
|T ∩ S| ≥ d for all S ∈ S(G ).
B. Ries ([email protected])
LAMSADE 2014
42 / 56
DEFINITIONS
Minimum d-transversal
Let G = (V , E ) be a weighted graph. For every vertex v ∈ V , consider a
cost c(v ) ≥ 0. Let d ≥ 1 be an integer.
A d-transversal T w.r. to S(G ) is a set T ⊆ V such that
|T ∩ S| ≥ d for all S ∈ S(G ).
The costPof a d-transversal T w.r. to S(G ) is defined as
c(T ) = v ∈T c(v ).
B. Ries ([email protected])
LAMSADE 2014
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DEFINITIONS
Minimum d-transversal
Let G = (V , E ) be a weighted graph. For every vertex v ∈ V , consider a
cost c(v ) ≥ 0. Let d ≥ 1 be an integer.
A d-transversal T w.r. to S(G ) is a set T ⊆ V such that
|T ∩ S| ≥ d for all S ∈ S(G ).
The costPof a d-transversal T w.r. to S(G ) is defined as
c(T ) = v ∈T c(v ).
A minimum d-transversal T w.r. to S(G ) is a d -transversal of
minimum total cost.
B. Ries ([email protected])
LAMSADE 2014
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DEFINITIONS
Minimum d-transversal
Let G = (V , E ) be a weighted graph. For every vertex v ∈ V , consider a
cost c(v ) ≥ 0. Let d ≥ 1 be an integer.
A d-transversal T w.r. to S(G ) is a set T ⊆ V such that
|T ∩ S| ≥ d for all S ∈ S(G ).
The costPof a d-transversal T w.r. to S(G ) is defined as
c(T ) = v ∈T c(v ).
A minimum d-transversal T w.r. to S(G ) is a d -transversal of
minimum total cost.
A minimum d -transversal T w.r. to S(G ) is called proper if for every
v ∈ T , T \ {v } is not a d -transversal w.r. to S(G ).
B. Ries ([email protected])
LAMSADE 2014
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PROBLEM
We are interested in the following problem:
B. Ries ([email protected])
LAMSADE 2014
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PROBLEM
We are interested in the following problem:
STAB-TRANS
Input: A weighted graph G = (V , E ), a nonnegative cost function c for V
and an integer d ≥ 1.
Output: A proper minimum d -transversal T w.r. to S(G ).
B. Ries ([email protected])
LAMSADE 2014
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COMPLEXITY RESULT
The following results concern graphs G = (V , E ) with w (v ) = c(v ) = 1 for
all v ∈ V .
B. Ries ([email protected])
LAMSADE 2014
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COMPLEXITY RESULT
The following results concern graphs G = (V , E ) with w (v ) = c(v ) = 1 for
all v ∈ V .
Theorem
For every fixed d ≥ 1, STAB-TRANS is NP-complete when G is the line
graph of a bipartite graph.
B. Ries ([email protected])
LAMSADE 2014
44 / 56
COMPLEXITY RESULT
The following results concern graphs G = (V , E ) with w (v ) = c(v ) = 1 for
all v ∈ V .
Theorem
For every fixed d ≥ 1, STAB-TRANS is NP-complete when G is the line
graph of a bipartite graph.
Theorem
Let G = (V , E ) be a bipartite graph. Then STAB-TRANS is
polynomial-time solvable.
B. Ries ([email protected])
LAMSADE 2014
44 / 56
COMPLEXITY RESULT
The following results concern graphs G = (V , E ) with w (v ) = c(v ) = 1 for
all v ∈ V .
Theorem
For every fixed d ≥ 1, STAB-TRANS is NP-complete when G is the line
graph of a bipartite graph.
Theorem
Let G = (V , E ) be a bipartite graph. Then STAB-TRANS is
polynomial-time solvable.
In what follows, we will consider STAB-TRANS in weighted bipartite
graphs with nonnegative cost function c.
B. Ries ([email protected])
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HYPOTHESES
Let G = (B, W, E ) be a weighted bipartite graph and let c be a
nonnegative cost function for V = B ∪ W.
B. Ries ([email protected])
LAMSADE 2014
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HYPOTHESES
Let G = (B, W, E ) be a weighted bipartite graph and let c be a
nonnegative cost function for V = B ∪ W.
all weights w (v ) satisfy w(v) > 0, ∀v ∈ V;
B. Ries ([email protected])
LAMSADE 2014
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HYPOTHESES
Let G = (B, W, E ) be a weighted bipartite graph and let c be a
nonnegative cost function for V = B ∪ W.
all weights w (v ) satisfy w(v) > 0, ∀v ∈ V;
all vertices in G are free.
B. Ries ([email protected])
LAMSADE 2014
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WEIGHTED STABLE SETS IN BIPARTITE
GRAPHS
Using network flows, we are able to show the following:
B. Ries ([email protected])
LAMSADE 2014
46 / 56
WEIGHTED STABLE SETS IN BIPARTITE
GRAPHS
Using network flows, we are able to show the following:
Theorem
Let G = (B, W, E ) be a weighted bipartite graph containing only free
vertices. There exists an unique partition V = (V1 , . . . , Vq ) of B ∪ W such
that a set S ⊆ B ∪ W is a maximum-weight stable set if and only if S is a
stable set and for any j ∈ {1, . . . , q} either S ∩ Vj = B ∩ Vj or
S ∩ Vj = W ∩ Vj .
B. Ries ([email protected])
LAMSADE 2014
46 / 56
WEIGHTED STABLE SETS IN BIPARTITE
GRAPHS
Using network flows, we are able to show the following:
Theorem
Let G = (B, W, E ) be a weighted bipartite graph containing only free
vertices. There exists an unique partition V = (V1 , . . . , Vq ) of B ∪ W such
that a set S ⊆ B ∪ W is a maximum-weight stable set if and only if S is a
stable set and for any j ∈ {1, . . . , q} either S ∩ Vj = B ∩ Vj or
S ∩ Vj = W ∩ Vj .
Remark
Since B and W are disjoint maximum-weight stable sets, we deduce that,
for any d ≥ 1, a d -transversal T w.r. to S(G ) must satisfy |T ∩ W| ≥ d
and |T ∩ B| ≥ d , and hence |T | ≥ 2d .
B. Ries ([email protected])
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PARTITION V OF V
V1
V3
V2
V4
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ADDITIONAL NOTIONS
From the partition V of V we define the following auxiliary digraph
G ∗ = (V ∗ , A∗ ):
B. Ries ([email protected])
LAMSADE 2014
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ADDITIONAL NOTIONS
From the partition V of V we define the following auxiliary digraph
G ∗ = (V ∗ , A∗ ):
V ∗ = {V1 , . . . , Vq } ;
B. Ries ([email protected])
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ADDITIONAL NOTIONS
From the partition V of V we define the following auxiliary digraph
G ∗ = (V ∗ , A∗ ):
V ∗ = {V1 , . . . , Vq } ;
A∗ = {(Vi , Vj )| ∃uv ∈ E , u ∈ Vi ∩ B, v ∈ Vj ∩ W i 6= j}.
B. Ries ([email protected])
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ADDITIONAL NOTIONS
From the partition V of V we define the following auxiliary digraph
G ∗ = (V ∗ , A∗ ):
V ∗ = {V1 , . . . , Vq } ;
A∗ = {(Vi , Vj )| ∃uv ∈ E , u ∈ Vi ∩ B, v ∈ Vj ∩ W i 6= j}.
Remark
It follows from the previous Theorem and from the definition of G ∗ that if
S ∈ S(G ) and if Vi is black w.r. to S, then all successors of Vi in G ∗ are
black w.r. to S.
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PARTITION V OF V
V1
V3
V1
V3
V2
V4
V2
V4
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G∗
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ADDITIONAL NOTIONS
From G ∗ we define the following relation:
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ADDITIONAL NOTIONS
From G ∗ we define the following relation:
Definition
u v if and only if
u ∈ W,v ∈ B
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ADDITIONAL NOTIONS
From G ∗ we define the following relation:
Definition
u v if and only if
u ∈ W,v ∈ B
u, v ∈ Vi or
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ADDITIONAL NOTIONS
From G ∗ we define the following relation:
Definition
u v if and only if
u ∈ W,v ∈ B
u, v ∈ Vi or
u ∈ Vi , v ∈ Vj , i 6= j, a directed path from Vi to Vj in G ∗
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ADDITIONAL NOTIONS
From G ∗ we define the following relation:
Definition
u v if and only if
u ∈ W,v ∈ B
u, v ∈ Vi or
u ∈ Vi , v ∈ Vj , i 6= j, a directed path from Vi to Vj in G ∗
Remark
It follows from the previous Theorem and from the previous Remark that if
u v , then for any maximum-weight stable set S we have S ∩ {u, v } =
6 ∅.
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RELATION V1
V1
V3
V3
v
V2
V4
V2
G∗
V4
u
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uv
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PROPER d-TRANSVERSALS
Lemma
Let G = (B, W, E ) be a weighted bipartite graph containing only free
vertices and let d ≥ 1 be an integer. Then T ⊆ V is a proper d -transversal
if and only if T is a set of d disjoint pairs of vertices (u, v ) such that u v .
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d-TRANSVERSAL ↔ MATCHING
Consider the bipartite graph G̃ = (B, W, Ẽ ) with vertex set B ∪ W and
edge set Ẽ = {uv : u ∈ W, v ∈ B, u v } and assign to each edge
uv ∈ Ẽ the cost c(uv ) = c(u) + c(v ).
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d-TRANSVERSAL ↔ MATCHING
Consider the bipartite graph G̃ = (B, W, Ẽ ) with vertex set B ∪ W and
edge set Ẽ = {uv : u ∈ W, v ∈ B, u v } and assign to each edge
uv ∈ Ẽ the cost c(uv ) = c(u) + c(v ).
v1
v1
V1
v3
v2
v10
v11
v4
V2
v5
v6
v9
v12
v2
v6
v3
v8
v4
v10
v7
v11
v9
v12
v14
V4
v7
v8
v5
V3
v13
v14
v13
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G̃
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MINIMUM d-TRANSVERSALS
Theorem
STAB-TRANS is polynomial-time solvable in weighted bipartite graphs.
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MINIMUM d-TRANSVERSALS
Theorem
STAB-TRANS is polynomial-time solvable in weighted bipartite graphs.
1
determine the sets of forced, excluded and free vertices
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MINIMUM d-TRANSVERSALS
Theorem
STAB-TRANS is polynomial-time solvable in weighted bipartite graphs.
1
determine the sets of forced, excluded and free vertices
2
delete forced and excluded vertices from G
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MINIMUM d-TRANSVERSALS
Theorem
STAB-TRANS is polynomial-time solvable in weighted bipartite graphs.
1
determine the sets of forced, excluded and free vertices
2
delete forced and excluded vertices from G
3
~
build G∗ and G
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MINIMUM d-TRANSVERSALS
Theorem
STAB-TRANS is polynomial-time solvable in weighted bipartite graphs.
1
determine the sets of forced, excluded and free vertices
2
delete forced and excluded vertices from G
3
~
build G∗ and G
4
add |f | independent edges e1 , . . . , e|f | to G̃
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MINIMUM d-TRANSVERSALS
Theorem
STAB-TRANS is polynomial-time solvable in weighted bipartite graphs.
1
determine the sets of forced, excluded and free vertices
2
delete forced and excluded vertices from G
3
~
build G∗ and G
4
add |f | independent edges e1 , . . . , e|f | to G̃
5
set c(ei ) = c(fi ), where fi is a forced vertex
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MINIMUM d-TRANSVERSALS
Theorem
STAB-TRANS is polynomial-time solvable in weighted bipartite graphs.
1
determine the sets of forced, excluded and free vertices
2
delete forced and excluded vertices from G
3
~
build G∗ and G
4
add |f | independent edges e1 , . . . , e|f | to G̃
5
set c(ei ) = c(fi ), where fi is a forced vertex
6
compute a minimum cost matching of size d
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CONCLUSION AND OPEN PROBLEMS
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CONCLUSION AND OPEN PROBLEMS
We presented some graph modification problems related to graph
parameters.
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CONCLUSION AND OPEN PROBLEMS
We presented some graph modification problems related to graph
parameters.
What about CONTRACTION BLOCKER(α) for interval graphs?
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CONCLUSION AND OPEN PROBLEMS
We presented some graph modification problems related to graph
parameters.
What about CONTRACTION BLOCKER(α) for interval graphs?
What about d -EDGE ADDITION BLOCKER(α) for bipartite graphs
for d ≥ 2?
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Thank you for your attention!
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