SPN graphs: When copositive = SPN

A real symmetric matrix A is copositive if xTAx≥0 for every nonnegative vector x. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative one. Every SPN matrix is copositive, but the converse does not hold for matrices of order greater than 4. A graph G is an SPN graph if every copositive matrix whose graph is G is SPN. In this paper we present sufficient conditions for a graph to be SPN (in terms of its possible blocks) and necessary conditions for a graph to be SPN (in terms of forbidden subgraphs). We also discuss the remaining gap between these two sets of conditions, and make a conjecture regarding the complete characterization of SPN graphs.