Conditional Matching Preclusion For The Star Graphs

Abstract

The matching preclusion number of an even graph G, denoted by mp(G), is the minimum number of edges whose deletion leaves the resulting graph without perfect matchings. The conditional matching preclusion number of an even graph G, denoted by mp1(G), is the minimum number of edges whose deletion leaves the resulting graph with neither perfect matchings nor isolated vertices. The class of (n,k)-star graphs is a popular class of interconnection networks for which the matching preclusion number and the classification of the corresponding optimal solutions were known. However, the conditional version of this problem was open. In this paper, we determine the conditional matching preclusion for (n,k)-star graphs as well as classify the corresponding optimal solutions via several new results. In addition, an alternate proof of the results on the matching preclusion problem will also be given.