Fractional matching preclusion number of graphs and the perfect matching polytope

Ruizhi Lin () and Heping Zhang ()
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Ruizhi Lin: Lanzhou University
Heping Zhang: Lanzhou University

Journal of Combinatorial Optimization, 2020, vol. 39, issue 3, No 15, 915-932

Abstract: 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 $$mp_f(G)$$mpf(G). We show $$mp_f(G)$$mpf(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 $$mp_f(G)$$mpf(G). For bipartite graph G, we obtain an explicit formula for $$mp_f(G)$$mpf(G) and show that $$\lfloor mp_f(G) \rfloor $$⌊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 $$mp_f(G \square H) \geqslant mp_f(G)+\lfloor mp_f(H) \rfloor $$mpf(G□H)⩾mpf(G)+⌊mpf(H)⌋, where $$G \square H$$G□H is the Cartesian product of G and H.

Keywords: Graph; Perfect matching; Matching preclusion; Linear program; Perfect matching polytope; Flow; 05C70; 05C72; 90C27; 90C35; 90C57 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-020-00530-2

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