Abstract

The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as König-Egerváry graphs. König-Egerváry graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding maximumKönig-Egerváry subgraphs of a given graph. More specifically, we look at the problem of finding a minimum number of vertices or edges to delete to make the resulting graph König-Egerváry. We show that both these versions are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. En route, we point out an interesting connection between the vertex deletion version and the A G V C problem where one is interested in the parameterized complexity of the V C problem when parameterized by the ‘additional number of vertices’ needed beyond the matching size. This connection is of independent interest and could be useful in establishing the parameterized complexity of A G V C problem.