Fractional matching preclusion number of graphs

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 $H$.