Maximum k -regular induced subgraphs

Independent sets, induced matchings and cliques are examples of regular induced subgraphs in a graph. In this paper, we prove that finding a maximum cardinality k-regular induced subgraph is an NP-hard problem for any value of k. We propose a convex quadratic upper bound on the size of a k-regular induced subgraph and characterize those graphs for which this bound is attained. Finally, we extend the Hoffman bound on the size of a maximum 0-regular subgraph (the independence number) from k = 0 to larger values of k. Acknowledgements: The first author’s research was supported by Centre for Research on Optimization and Control (CEOC) from the ”Fundação para a Ciência e a Tecnologia” FCT, cofinanced by the European Community Fund FEDER/POCTI.