Fractional matching preclusion number of graphs and the perfect matching polytope

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 program which can be used to find the matching preclusion number of graphs. In this paper, by relaxing of the 0-1 linear program we obtain a linear program and call its optimal objective value as fractional matching preclusion number of graph G, denoted by mpf (G). We showmpf (G) can be computed in polynomial time for any graph G. By using the perfect matching polytope, we transform it into a new linear program whose optimal value equals the reciprocal of mpf (G). For bipartite graph G, we obtain an explicit formula for mpf (G) and show that mpf (G) is the maximum integer k such that G has a k-factor. Moreover, for any two bipartite graphs G and H, we show mpf (G H) mpf (G) + mpf (H), where G H is the Cartesian product of G and H.