  A real symmetric matrix $A$ is copositive if $x^TAx\ge 0$ 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.
