Integer k-matching preclusion of graphs

TL;DR

This paper introduces the integer k-matching preclusion number as a generalization of matching preclusion, analyzing its properties and calculating it for various classes of graphs.

Contribution

It defines the (strong) integer k-matching preclusion number and establishes its relationship with fractional matching preclusion, providing formulas for specific graph classes.

Findings

When k is even, MP^k equals the fractional matching preclusion number.

Necessary conditions for graphs with almost-perfect integer k-matching.

Explicit calculations of MP^k and SMP^k for complete, bipartite, and arrangement graphs.

Abstract

As a generalization of matching preclusion number of a graph, we provide the (strong) integer $k$-matching preclusion number, abbreviated as $MP^{k}$ number ($SMP^{k}$ number), which is the minimum number of edges (vertices and edges) whose deletion results in a graph that has neither perfect integer $k$-matching nor almost perfect integer $k$-matching. In this paper, we show that when $k$ is even, the ($SMP^{k}$) $MP^{k}$ number is equal to the (strong) fractional matching preclusion number. We obtain a necessary condition of graphs with an almost-perfect integer $k$-matching and a relational expression between the matching number and the integer $k$-matching number of bipartite graphs. Thus the $MP^{k}$ number and the $SMP^{k}$ number of complete graphs, bipartite graphs and arrangement graphs are obtained, respectively.