A note on generalized matching preclusion in bipartite graphs

Abstract

For a graph with an even number of vertices, the matching preclusion number is the minimum number of edges whose deletion results in a graph with no perfect matchings. The conditional matching preclusion number, introduced as an extension of the matching preclusion number, has the additional requirement that the resulting graph have no isolated vertices. In bipartite graphs, the edge sets that achieve these minimum numbers can be construed in terms of obstruction sets of vertices. From this perspective, the generalized matching preclusion problem was developed; the matching preclusion number of order a is the minimum number of edges whose deletion results in a graph with no perfect matchings and no obstruction sets with fewer than a vertices. In this paper, we extend known sufficient conditions regarding the matching preclusion and conditional matching preclusion numbers (orders 1 and 2, respectively) to give sufficient conditions for the matching preclusion problem of order a, for any positive integer a.