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Extendability and Criticality in Matching Theory

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Abstract

A connected graph G has property E(m, n) (or more briefly “G is E(m, n)”) 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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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand

    R. E. L. Aldred

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Michael D. Plummer

Authors
  1. R. E. L. Aldred
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  2. Michael D. Plummer
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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 36, 573–589 (2020). https://doi.org/10.1007/s00373-020-02139-y

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