1. Clique-Width of Monogenic Bipartite Graphs Jordan Volz DIMACS REU 2006 Mentor: Dr. Vadim Lozin, RUTCOR

2. Overview Provide definitions Present problem Provide necessary tools Outline Solution

3. 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.

4. 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  i,j : Join all vertices of label i to label j  i  j : re-label all vertices of label i to label j We say a family of graphs F has bounded clique- width if there exists an integer k such that for any graph G in F, G can be constructed using k labels. If no such k exists, F has unbounded clique- width. Clique-Width

5.   Clique-Width Example t 1 =  r  b [  b,r (  w,b (w(1)+b(2))+  w,r (w(5) + r(6)))] t 2 =  b  r [  b,r (  w,b (w(4)+b(3))+  w,r (w(8) + r(7)))]  b,r (t 2 + t 2 ) Clique Width is at most 3.     

6. 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 K 1,3 -free and 2P 3 -free graphs.

7. Forbidden Induced Subgraphs Lozin and Rautenbach described 8 graphs in S that are self- complementary. Previously results were known for A 1 and A 2 (bounded), as well as A 3 and A 5 together. We will resolve the case for A 3, A 5, A 6, and A 8.        

8. Tools for Proof of K 1,3 +e 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 x 1,…x n such that: N {x 2,…x n } (x 1 )={x 2 } or N {x 2,…x n } (x 1 )= {x 4,…x n }. For 1<i<n, if N {x i,…x n } (x i-1 )={x i } then N {x i+1,…x n } (x i )= {x i+3,…x n } and if N {x i,…x n } (x i-1 )= {x i+3,…x n } then N {x i+1,…x n } (x i )= {x i+1 }. Clique-Width of a well orderable graph is at most 5. K S

9. Outline of Proof for K 1,3 +e We know that the class of S 1,2,3 -free have bounded clique width (Lozin 2002). But, S 1,2,3 is H 7 ! Thus we conclude that a K 1,3 +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 H p, 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

10. Tools for Proof of 2P 3 Bipartite Chain Graph: A bipartite graph with vertex ordering b 1,…,b n and w 1,…,w, m such that N(b i ) is contained in N(b i+1 ) (N(w i ) is contained in N(w i-1 )). Bipartite chain graphs have bounded clique width, and even multi-layered bipartite chain graphs have bounded clique width. But, 2P 3 -free graphs have unbounded clique-width!

11. 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. 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 2P 3 free graph that has unbounded clique-width, and then that characterizes the entire class.

12. Outline of Proof for 2P 3 Brandstadt et al (2003) proved that (K 4, 2K 2 )-free graphs have unbouded clique-width by using the following type of graph: We observe that the following graph is 2P 3 free:

13. Outline of Proof for 2P 3 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. Since A 6 and A 8 contain A 3, we also conclude that this exact same graph works to show their respective graph classes have unbounded clique width as well.

14. Open Questions Prove unbounded clique width of A 4 -free graphs (and hence A 7 ) Prove bounded clique width of A 7 -free graphs (and hence A 4 ) Prove bounded clique width of A 4 -free graphs and unbounded clique width of A 4 -free graphs. More info: dimax.rutgers.edu/~jordanv