Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs

Abstract

A homogeneous set of a graph G is a set X of vertices such that 2 ≤ |X| < |V (G)| and no vertex in V (G) − X has both a neighbor and a non-neighbor in X. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.