Article Title

SPN Graphs


Copositive, SPN, F_5, fan, completion, matrix, graph


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 [N. Shaked-Monderer. SPN graphs: When copositive = SPN. Linear Algebra Appl., 509:82{113, 2016.], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. This conjecture is disproved, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K_4 is a SPN graph if and only if at most one edge is subdivided. It is conjectured that a graph is an SPN graph if and only if it does not have an F_5 minor, where F_5 is the fan on five vertices. To establish that the complete subdivision graph of K_4 is not an SPN graph, rank-1 completions are introduced and graphs that are rank-1 completable are characterized.

abs_vol35_pp376-386.pdf (116 kB)
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