Conditional matching preclusion sets

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. In this paper, we look for obstruction sets beyond these sets. We introduce the conditional matching preclusion number of a graph. It is the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. We find this number and classify all optimal sets for several basic classes of graphs.