A simplex like approach based on star sets for recognizing convex-QP adverse graphs

A graph G with convex-QP stability number (or simply a convex-QP graph) is a graph for which the stability number is equal to the optimal value of a convex quadratic program, say P (G). There are polynomial-time procedures to recognize convex-QP graphs, except when the graph G is adverse or contains an adverse subgraph (that is, a non complete graph, without isolated vertices, such that the least eigenvalue of its adjacency matrix and the optimal value of P (G) are both integer and none of them changes when the neighborhood of any vertex of G is deleted). In this paper, from a characterization of convex-QP graphs based on star sets associated to the least eigenvalue of its adjacency matrix, a simplex-like algorithm for the recognition of convex-QP adverse graphs is introduced.