Conditional matching preclusion for the alternating group graphs and split-stars

The matching preclusion number of a graph is the minimum number of edges the deletion of which 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. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges the deletion of which results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this article, we find this number and classify all optimal sets for the alternating group graphs, one of the most popular interconnection networks, and their companion graphs, the split-stars. Moreover, some general results on the conditional matching preclusion problems are also presented.