Fractional matching preclusion number of graphs

TL;DR

This paper introduces the fractional matching preclusion number of graphs, a polynomial-time computable relaxation of the matching preclusion number, with explicit formulas for bipartite graphs and properties under graph products.

Contribution

It defines the fractional matching preclusion number via linear programming relaxation, provides explicit formulas for bipartite graphs, and explores its behavior under Cartesian products.

Findings

mp_f(G) can be computed in polynomial time

Explicit formula for bipartite graphs involving k-factors

Inequality for mp_f of Cartesian product of bipartite graphs

Abstract

Let $G$ be a graph with an even number of vertices. The matching preclusion number of $G$, denoted by $mp(G)$, is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching. We introduced a $0$-$1$ linear programming which can be used to find matching preclusion number of graphs. In this paper, by relaxing of the $0$-$1$ linear programming we obtain a linear programming and call its optimal objective value as fractional matching preclusion number of graph $G$, denoted by $mp_f(G)$. We show $mp_f(G)$ can be computed in polynomial time for any graph $G$. By using perfect matching polytope, we transform it as a new linear programming whose optimal value equals the reciprocal of $mp_f(G)$. For bipartite graph $G$, we obtain an explicit formula for $mp_f(G)$ and show that $\lfloor mp_f(G) \rfloor$ is the maximum integer $k$ such that $G$ has a $k$-factor. Moreover, for any two bipartite graphs $G$ and $H$, we show $mp_f(G \square H) \geqslant mp_f(G)+\lfloor mp_f(H) \rfloor$, where $G \square H$ is the Cartesian product of $G$ and $H$.