Journal of Combinatorial Optimization

, Volume 31, Issue 1, pp 311–326 | Cite as

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

  • Domingos M. Cardoso
  • Carlos J. LuzEmail author


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.


Convex quadratic programming in graphs Star sets Graphs with convex-\(QP\) stability number Simplex-like approach 



The authors thank the two referees for their helpful comments and suggestions that improved the paper. The research of both authors is partially supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal

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