Abstract
We investigate the ratio \(\mathcal {I}(G)\) 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 \(\nu (G)\), this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, \(\mathcal {I}(G)\) approaches \(\frac{1}{2}\). We propose a general technique to determine the asymptotic behavior of \(\mathcal {I}(G)\) 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 \(\mathcal {I}(G)\) which were typically obtained using generating functions, and we then determine the asymptotic value of \(\mathcal {I}(G)\) for other families of graphs, highlighting the spectrum of possible values of this graph invariant between \(\frac{1}{2}\) and 1.
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References
Andriantiana EOD, Razanajatovo Misanantenaina V, Wagner S (2020) The average size of matchings in graphs. Graphs Comb 36(3):539–560
Arnosti N (2021) Greedy matching in bipartite random graphs. Stochastic Syst 12:133
Aronson J, Dyer M, Frieze A, Suen S (1994) On the greedy heuristic for matchings. In: Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (USA, 1994), SODA ’94, Society for Industrial and Applied Mathematics, pp. 141–149
Aronson J, Dyer M, Frieze A, Suen S (1995) Randomized greedy matching. ii. Random Struct. Algorithms 6(1):55–73
Besser B, Poloczek M (2017) Greedy matching: guarantees and limitations. Algorithmica 77(1):201–234
Devillez G, Hauweele P, Mélot H (2019) PHOEG Helps to Obtain Extremal Graphs. In: Operations Research Proceedings 2018 (GOR (Gesellschaft fuer Operations Research e.V.)) (sept. 12-14 2019), Fortz B, Labbé M, Eds., Springer, Cham, p. 251 (Paper 32)
Diestel R (2017) Graph theory, 2nd edn. Springer-Verlag, Berlin
Došlić T, Short T (2021) Maximal matchings in polyspiro and benzenoid chains. Appl Anal Discr Math 15(1):179–200
Došlić T, Zubac I (2015) Counting maximal matchings in linear polymers. Ars Math Contemp 11(2):255–276
Dyer M, Frieze A (1991) Randomized greedy matching. Random Struct Algorithms 2(1):29–45
Dyer M, Frieze A, Pittel B (1993) The average performance of the greedy matching algorithm. Ann Appl Probab 3:526–552
Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press, Cambridge
Flory PJ (1939) Intramolecular reaction between neighboring substituents of vinyl polymers. J Am Chem Soc 61(6):1518–1521
Frieze A, Radcliffe AJ, Suen S (1995) Analysis of a simple greedy matching algorithm on random cubic graphs. Comb Probab Comput 4(1):47–66
Goel G, Tripathi P (2012) Matching with our eyes closed. In: 2012 IEEE 53rd annual symposium on foundations of computer science, IEEE, pp. 718–727
Gutman I (1998) Distance of thorny graphs. Publ l’Instit Math 63(77):31–36
Hosoya H (1971) Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull Chem Soc Jpn 44:2332–2339
Huntemann S, McKay NA (2021) Counting domineering positions. J Integer Seq 24(4):8
Jackson JL, Montroll EW (1958) Free radical statistics. J Chem Phys 28(6):1101–1109
Lovász L, Plummer MD (2009) Matching theory, vol 367. American Mathematical Soc, Providence
Magun J (1997) Greedy matching algorithms, an experimental study. In: Proceedings of the 1st workshop on algorithm engineering. 6:22–31
Miller Z, Pritikin D (1997) On randomized greedy matchings. Random Struct Algorithms 10(3):353–383
Poloczek M, Szegedy M (2012) Randomized greedy algorithms for the maximum matching problem with new analysis. In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, IEEE, pp. 708–717
Short T, Ash Z (2019) The number of maximal matchings in polyphenylene chains. Iran J Math Chem 10(4):343–360
Tinhofer G (1984) A probabilistic analysis of some greedy cardinality matching algorithms. Ann Oper Res 1(3):239–254
Van der Laan H (1997) Le nombre plastique. Brill Archive, Leiden
Yannakakis M, Gavril F (1980) Edge dominating sets in graphs. SIAM J Appl Math 38(3):364–372
Acknowledgements
The authors would like to thank Richard Labib for his help, an anonymous referee for the elegant proof of Theorem 1, and another anonymous referee for useful chemical references.
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Hertz, A., Bonte, S., Devillez, G. et al. The average size of maximal matchings in graphs. J Comb Optim 47, 46 (2024). https://doi.org/10.1007/s10878-024-01144-8
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DOI: https://doi.org/10.1007/s10878-024-01144-8