On the Clique-Width of Graphs with Few P4's

Linear time solv-able optimization problems on certain structured graph families, extended abstract. On the xed parameter complexity of graph enumeration problems deenable in monadic second order logic. in preparartion, 1998. 35 the rst case there is no other vertex z of L such that w and z occur in the same equivalence class of R red , or else z and w are adjacent to both u and v, a contradiction. In the second case there is no vertex z of L ? P u such that z and a vertex of P u occur in the same equivalence class of R red , or else u must be adjacent to z, a contradiction. We conclude that for each vertex of S 3;blue , or for each pair of vertices of S 3;blue , there is a corresponding vertex of L or a corresponding set of vertices of L, which occur in an equivalence class of R red which is diierent from all the equivalence classes in which the other vertices of L occur. It follows the vertices of L occur in at least jS 3;blue j=2 n=72 diierent equivalence classes of R red. But this contradicts the assumption that the number of equivalence classes of R red is less than n=72. 2 Theorem 4. There is a class C containing innnitely many split graphs, such that for every graph G = hV; Ei 2 C cwd(G) (sqrt2jV j ? 1)=72. Proof: Let G n denote the clique of n vertices, and let C be the class of split graphs of cliques deened by: C = fsplt(G n) : n 2 N; n 20g. From Theorem 5 and Lemma 4 above it follows that for n 2 N n 20, cwd(splt(G n)) n=72. Since for every clique G n splt(G n) is a split graph, it follows that the class C satisses the conditions of the theorem. 2 Acknowledgments We are indebted to Luitpold Babel who made us aware of the (q; q ? 3) graphs and the prime graphs associated with their corresponding modular decompositions. References Bab98a] L. Babel. On graphs with simple P 4 {structure. Preprint, 1998. 34 CASE 1: Suppose that either jS 1 j < 10n=18 or jS 2 j < 10n=18. We assume that jS 1 j < 10n=18, (the case when jS 2 j < 10n=18 can be handled similarly). From the deenition of …