Clique Width of Monogenic
Bipartite Graphs
Jordan Volz
DIMACS REU 2006
Mentor: Dr. Vadim Lozin, RUTCOR
Overview
Provide definitions
Present problem
Discuss progress made
Introduce research problem
Definitions
Graph: A graph G is defined to be a set of
vertices V(G) and a set of edges E(G). If
x=(a,b), where a and b are vertices of G, we
say a is adjacent to b.
Bipartite Graph: A Bipartite Graph is a graph
G whose vertices can be partition into two sets
W and B where vertices of one set are adjacent
to only vertices of the other set.
Induced Subgraph: If A is a subset of V(G),
then the graph formed by the vertices of A and
the edges between them is an induced
subgraph of G.
Definitions
Forbidden Subgraph: A graph H that
cannot appear as an induced subgraph in G
Monogenic Class: A class of graphs defined
by a single forbidden subgraph H.
Clique Width: The minimum number of
labels needed to construct a graph G using
the following 4 operations:




i(v): Creation of a new vertex v with label I
G + H: Disjoint union of two labeled graphs
hi,j: Join all vertices of label i to label j
rigj: re-label all vertices of label i to label j
If a graph G can be constructed using k labels,
the algebraic expression used to define G is called
a k-expression and G has bounded clique width.
Clique Width Example
t1=rrgb[hb,r(hw,b(w(1)+b(2))+hw,r(w(5) + r(6)))]
t2=rbgr[hb,r(hw,b(w(4)+b(3))+hw,r(w(8) + r(7)))]
hb,r(t2 + t2)
1
2
3
4
5
6
Clique Width is at most 3.
7
8
P vs NP
P: class of decision problems that can be
solved in polynomial time relative to input
size.
NP: class of decision problems whose
solutions can be checked in polynomial time.
NP-hard: class of decision problems in NP
such that for any decision problem in NP
there exists a polynomial time reduction to a
problem in NP-hard. (most likely not to be P).
NP
$ x 1,000,000
?
NP-hard
P
P=NP
Why Study Clique Width?
In general, there is no known method to solve NPhard problems in polynomial time.
Some graph problems have polynomial time solutions
when restrictions are placed upon graph structure.
Many NP-hard problems have polynomial-time
solutions restricted to graphs of bounded clique width
(Courcelle, Engelfriet, Rozenberg, 1993).
We study (bounded) clique width to find graphs that
are NP-hard in general but have linear time solutions
in special cases.
Bipartite Graphs and Clique Width
Bipartite graphs have unbounded clique width
i
in the general case (grids, permutation graphs).
Important Result: Let S be the class of graphs
where every connected component is in the
form Si,j,k. If a class X of bipartite graphs
defined by a set F of forbidden induced
subgraphs is bounded then F contains a graph
in S and a graph the bipartite complement
which is in S. (Lozin, Rautenbach, 2006).
Also, bipartite graphs that are S1,2,3-free have
j
bounded clique width (Lozin 2002).
k
My Research
Lozin and Rautenbach described 8 graphs in S that
are self-complementary.
A1
A5
A2
A6
A3
A7
A4
A8
My Research
Lozin and Rautenbach also showed that forbidding A1
or A2 results in bounded clique width.
Vanherpe (2004) proved that forbidding A3 and A5
together results in bounded clique width.
I aim to prove that forbidding either A3 or A5 will
result in a bounded clique width, as well as looking at
other monogenic classes of graphs and clique width
on bipartite graphs in general.
Lozin’s results describes a necessary condition for a
graph to be bounded – is there a sufficient one?
My Research
Time permitting I will also look at bipartite
permutation graphs.
Permutation graphs are formed by taking two copies
of a graph G and joining each vertex in one copy to
exactly one vertex in another defined by a
permutation p (vi is adjacent to vp(i)).
Bipartite permutation graphs are interesting due to
the following fact: clique width of bipartite
permutation graphs is unbounded, yet some NP-hard
problems have efficient solutions when restricted to
bipartite permutation graphs.
The goal is to discover the characteristic of bipartite
permutation graphs that yields efficient solutions for
hard problems. Obviously this is something more
general than clique width.