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

2. Overview Provide definitions Present problem Discuss progress made Introduce research problem

3. 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.

4. 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  i,j : Join all vertices of label i to label j  i  j : re-label all vertices of label i to label j Definitions 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.

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. 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). P NP-hard NP P=NP ? $ x 1,000,000

7. Why Study Clique Width? In general, there is no known method to solve NP- hard 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.

8. Bipartite Graphs and Clique Width Bipartite graphs have unbounded clique width in the general case (grids, permutation graphs). Important Result: Let S be the class of graphs where every connected component is in the form S i,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 S 1,2,3 -free have bounded clique width (Lozin 2002). i j k

9. My Research Lozin and Rautenbach described 8 graphs in S that are self-complementary.        

10. My Research Lozin and Rautenbach also showed that forbidding A 1 or A 2 results in bounded clique width. Vanherpe (2004) proved that forbidding A 3 and A 5 together results in bounded clique width. I aim to prove that forbidding either A 3 or A 5 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?

11. 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  (v i is adjacent to v  (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.