König Deletion Sets and Vertex Covers above the Matching Size

A graph is König-Egerváry if the size of a minimum vertex cover equals the size of a maximum matching in the graph. We show that the problem of deleting at most k vertices to make a given graph König-Egerváry is fixedparameter tractable with respect to k. This is proved using interesting structural theorems on matchings and vertex covers which could be useful in other contexts. We also show an interesting parameter-preserving reduction from the vertexdeletion version of red/blue-split graphs [4, 9] to a version of V C and as a by-product obtain 1. the best-known exact algorithm for the optimization version of O C T [15]; 2. fixed-parameter algorithms for several vertex-deletion problems including the following: deleting k vertices to make a given graph (a) bipartite [17], (b) split [5], and (c) red/blue-split [7].