On the Relationship between Clique-Width and Treewidth

Abstract

Treewidth is generally regarded as one of the most useful parameterizations of a graph’s construction. Clique-width is a similar parameterizations 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, Courcelle 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.