# 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}.

## Keywords

Polynomial Time Algorithm Construction Tree Order Logic Interval Graph Parse Tree## Preview

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