Conditional Matching Preclusion Number of Graphs

Abstract

The conditional matching preclusion number of a graph $G$ , denoted by $m{p}_1\left(G\right)$ , is the minimum number of edges whose deletion results in the graph with no isolated vertices that has neither perfect matching nor almost-perfect matching. In this paper, we first give some sharp upper and lower bounds of conditional matching preclusion number. Next, the graphs with large and small conditional matching preclusion numbers are characterized, respectively. In the end, we investigate some extremal problems on conditional matching preclusion number.