On the Relationship between Clique-Width and Treewidth (Extended Abstract) Derek G. Corneil 1 and Udi Rotics 2 1 Department of Computer Science, University of Toronto, Toronto, Ontario, Canada 2 School of Mathematics and Computer Science, Netanya Academic College, Netanya, Israel Abstract. Treewidth is generally regarded as one of the most useful pa- rameterizations of a graph’s construction. Clique-width is a similar pa- rameterizations that shares one of the powerful properties of treewidth, namely: if a graph is of bounded treewidth (or clique-width), then there is a polynomial time algorithm for any graph problem expressible in Monadic Second Order Logic, using quantifiers on vertices (in the case of clique-width you must assume a clique-width parse expression is given). In studying the relationship between treewidth and clique-width, Cour- celle and Olariu showed that any graph of bounded treewidth is also of bounded clique-width; in particular, for any graph G with treewidth k, the clique-width of G ≤ 4 * 2 k-1 + 1. In this paper, we improve this result to the clique-width of G ≤ 3 * 2 k-1 and more importantly show that there is an exponential lower bound on this relationship. In particular, for any k, there is a graph G with treewidth = k where the clique-width of G ≥ 2 ⌊k/2⌋-1 . 1 Introduction One of the most fruitful graph theoretical developments of the last few decades hasbeentheconceptoftreewidth,pioneeredbyRobertsonandSeymour.Loosely speaking, the treewidth of a graph captures a way of constructing the graph in a “tree like” fashion. The lower the treewidth of a graph, the closer it is to being a tree (connected treewidth 1 graphs are precisely trees). One of the major results in this area is that any problem expressible in Monadic Second Order Logic (which includes many NP-complete graph problems) when restricted to graphs of bounded treewidth k, has a linear time algorithm (albeit with a constant that grows exponentially with k). Although this result is very far reaching, it is still somewhat dissatisfying since many classes of “tame” graphs, for example cliques, have arbitrarily high treewidth, yet have simple linear time algorithms for most of the problems mentioned above. The clique-width of a graph is another attempt to parameterize the con- struction of a graph so that sweeping claims can be made about the graph’s tractability for polynomial time solutions to difficult problems. A. Brandst¨adt and V.B. Le (Eds.): WG 2001, LNCS 2204, pp. 78–90, 2001. c  Springer-Verlag Berlin Heidelberg 2001