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Extendability and Criticality in Matching Theory

Abstract

A connected graph G has property E(mn) (or more briefly “G is E(mn)”) if for every pair of disjoint matchings M and N in G with \(|M|=m\) and \(|N|=n\) respectively, there is a perfect matching F in G such that \(M\subseteq F\) and \(N\cap F=\emptyset \). In particular, a graph which has the E(m, 0) property is said to be m-extendable. Also a graph G which has the property that \(G-u-v\) has a perfect matching for every pair of distinct vertices u and v is said to be bicritical. In the present paper we investigate further the interrelation between extendability and criticality. In Lovász and Plummer (Matching theory, North-Holland Publisher, Amsterdam, 1986) author proved that a 2-extendable graph is either 1-extendable and bipartite, or else bicritical (clearly a graph cannot be both). In the present paper we generalize this result by extending the notion of bicriticality in several ways and studying the relationships of these extensions to extendability.

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Correspondence to R. E. L. Aldred.

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Aldred, R.E.L., Plummer, M.D. Extendability and Criticality in Matching Theory. Graphs and Combinatorics (2020). https://doi.org/10.1007/s00373-020-02139-y

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Keywords

  • Matching extension
  • Bicritical
  • k-critical