The average size of maximal matchings in graphs

Alain Hertz (), Sébastien Bonte, Gauvain Devillez and Hadrien Mélot
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Alain Hertz: Polytechnique Montréal - Gerad
Sébastien Bonte: University of Mons
Gauvain Devillez: University of Mons
Hadrien Mélot: University of Mons

Journal of Combinatorial Optimization, 2024, vol. 47, issue 3, No 7, 34 pages

Abstract: Abstract We investigate the ratio $$\mathcal {I}(G)$$ I ( G ) of the average size of a maximal matching to the size of a maximum matching in a graph G. If many maximal matchings have a size close to $$\nu (G)$$ ν ( G ) , this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, $$\mathcal {I}(G)$$ I ( G ) approaches $$\frac{1}{2}$$ 1 2 . We propose a general technique to determine the asymptotic behavior of $$\mathcal {I}(G)$$ I ( G ) for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of $$\mathcal {I}(G)$$ I ( G ) which were typically obtained using generating functions, and we then determine the asymptotic value of $$\mathcal {I}(G)$$ I ( G ) for other families of graphs, highlighting the spectrum of possible values of this graph invariant between $$\frac{1}{2}$$ 1 2 and 1.

Keywords: Maximal matching; Average size; Graph invariant; Asymptotic value (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10878-024-01144-8

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