Quasi-brittle graphs, a new class of perfectly orderable graphs

Abstract

A graph G is quasi-brittle if every induced subgraph H of G contains a vertex which is incident to no edge extending symmetrically to a chordless path with three edges in either H or its complement H¯. The quasi-brittle graphs turn out to be a natural generalization of the well-known class of brittle graphs. We propose to show that the quasi-brittle graphs are perfectly orderable in the sense of Chvátal: there exists a linear order < on their set of vertices such that no induced path with vertices a, b, c, d and edges ab, bc, cd has a < b and d < c.