The Complexity of Finding Subgraphs Whose Matching Number Equals the Vertex Cover Number

The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as K¨onig-Egerv´ary graphs. K¨onig-Egerv´ary graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding maximum K¨onig-Egerv´ary 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 K¨onig-Egerv´ary. 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...

• Type: Conference
• Year: 2007
• Where: ISAAC
• Authors: Sounaka Mishra, Venkatesh Raman, Saket Saurabh, Somnath Sikdar, C. R. Subramanian
• Keywords: ISAAC 2007, K¨onig-Egerv´ary Graphs, Parameterized Complexity, V C Problem