Clique-Width of Monogenic Bipartite Graphs

Open Questions

Prove unbounded clique width of A4-free graphs (and hence A7) Prove bounded clique width of A7-free graphs (and hence A4)


Prove bounded clique width of A4-free graphs and unbounded clique width of A4-free graphs.
Definitions

Graph: Vertices and Edges Bipartite Graph: A graph 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. Forbidden Induced Subgraph: A graph H that cannot appear as an induced subgraph in G Monogenic Class: A class of graphs defined by a single forbidden induced subgraph H.
Unbounded clique-width

How do we show a graph has unbounded clique-width? To show bounded clique width, you need only to produce an algorithm to construct a k-expression.

Clique Width is at most 3.
Why Study Clique Width?

In general, there is no known method to solve NP-hard problems in polynomial time. Many NP-hard problems have polynomial-time solutions restricted to graphs of bounded clique width (Courcelle, Engelfriet, Rozenberg, 1993). Bipartite graphs have unbounded clique width in the general case (grids, permutation graphs). We’re interested in monogenic classes of bipartite graphs, specifically K1,3-free and 2P3-free graphs.

More info: /~jordanv
S

Clique-Width of a well orderable graph is at most 5.
Outline of Proof for K1,3 +e


We know that the class of S1,2,3-free have bounded clique width (Lozin 2002). But, S1,2,3 is H7 ! Thus we conclude that a K1,3 +e – free graph must have an induced subgraph that is a well-orderable graph of size at least 7. When G has a well orderable subgraph Hp , we then can break the problem down into 3 cases, when p>10, when p=8 or 9, and when p=7. In all three cases we use similar logic to show that G must have bounded clique width, using lots of



Forbidden Induced Subgraphs

Lozin and Rautenbach described 8 graphs in S that are selfcomplementary.
A1
A2
A3
A4
A5

A6
A7
A8
Previously results were known for A1 and A2 (bounded), as well as A3 and A5 together. We will resolve the case for A3, A5, A6, and A8.

Bipartite chain graphs have bounded clique width, and even multi-layered bipartite chain graphs have bounded clique width.

But, 2P3-free graphs have unbounded clique-width!

We observe that the following graph is 2P3 free:
Outline of Proof for 2P3

Then we find a set of n vertices that we split into three different groups based on their adjacencies in the graph. Each set we prove has pairwise different labels, so we conclude that this graph uses at least n/3 labels.
Tools for Proof of K1,3 +e

K
K+S Graph: a bipartite graph that can be partitioned into a biclique and an independent set. Well Orderable Graph: A bipartite graph whose vertex set can be ordered x1,…xn such that: N{x ,…x }(x1)={x2} or N {x ,…x }(x1)= {x4,…xn}. 2 n 2 n For 1<i<n, if N{x ,…x }(xi-1)={xi} then N {x (x )= {xi+3,…xn} i n i+1,…xn} i and if N{xi,…xn}(xi-1)= {xi+3,…xn} then N {xi+1,…xn}(xi)= {xi+1}.


There are very few proofs for showing graphs have unbounded clique width.
So we steal an existing proof and modify it to our needs. What we need is one example of a 2P3 free graph that has unbounded clique-width, and then that characterizes the entire class.


Clique-Width Example
ห้องสมุดไป่ตู้
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 7 8
Tools for Proof of 2P3

Bipartite Chain Graph: A bipartite graph with vertex ordering b1,…,bn and w1,…,w,m such that N(bi) is contained in N(bi+1) (N(wi) is contained in N(wi-1)).

Since A6 and A8 contain A3, we also conclude that this exact same graph works to show their respective graph classes have unbounded clique width as well.


Outline of Proof for 2P3

Brandstadt et al (2003) proved that (K4, 2K2)-free graphs have unbouded clique-width by using the following type of graph:
Clique-Width of Monogenic Bipartite Graphs
Jordan Volz DIMACS REU 2006 Mentor: Dr. Vadim Lozin, RUTCOR