On brittle graphs

Chvátal defined a graph G to be brittle if each induced subgraph F of G contains a vertex that is not a midpoint of any P₄ or not an endpoint of any P₄. Every brittle graph is perfectly orderable. In this paper, we prove that a graph is brittle whenever it is HHD-free (containing no chordless cycle with at least five vertices, no cycle on six vertices with a long chord, and no complement of the chordless path on five vertices). We also design an O(n₄) algorithm to recognize HHD-free graphs, and also an O(n₄) algorithm to construct a perfect order of an HHD-free graph. It follows from this result that an optimal coloring and a largest clique of an HHD-free graph can be found in O(n₄) time.