A simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [Shaked-Monderer, SPN graphs: When copositive = SPN, Linear Algebra Appl., 509(15):82-113, 2016], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. We disprove this conjecture, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K4 is a SPN graph if and only if at most one edge is subdivided. We conjecture that a graph is an SPN graph if and only if it does not have an F5 minor, where F5 is the fan on five vertices. To establish that the complete subdivision graph of K4 is not an SPN graph, we introduce rank-1 CP completions and characterize graphs that are rank-1 CP completable.
Hogben, Leslie and Shaked-Monderer, Naomi, "SPN graphs and rank-1 CP-completable graphs" (2018). Mathematics Publications. 209.