Minimal graphs for matching extensions.

Abstract : Given a positive integer n we find a graph G = (V, E) on |V | = n vertices with a minimum number of edges such that for any pair of vertices x, y with xy ∈ E the graph G−x−y contains a (almost) perfect matching M. Intuitively the non edge xy together with M form a (almost) perfect matching of G. Similarly we determine a graph G = (V, E) with a minimum number of edges such that for any matching ¯ M of the compement ¯ G of G with size n 2 − 1, G − V (¯ M) contains an edge e. Here ¯ M together with the edge e of G form a (almost) perfect matching of ¯ G. We characterize these minimum graphs for all values of n.
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Marie-Christine Costa, Dominique de Werra, C Picouleau. Minimal graphs for matching extensions.. Discrete Applied Mathematics, Elsevier, 2018, 234, pp.47-55. ⟨10.1016/j.dam.2015.11.007⟩. ⟨hal-01413150⟩

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