Fractional matching preclusion number of graphs and the perfect matching polytope

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 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 mp f (G). We show mp f (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 mp f (G). For bipartite graph G, we obtain an explicit formula for mp f (G) and show that ⌊ mp f (G) ⌋ 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 (Gs H) ⩾ mp f (G) + ⌊ mp f (H) ⌋ , where Gs H is the Cartesian product of G and H. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.