The average size of maximal matchings in graphs

IF 0.9 4区 数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS Journal of Combinatorial Optimization Pub Date : 2024-04-04 DOI:10.1007/s10878-024-01144-8
Alain Hertz, Sébastien Bonte, Gauvain Devillez, Hadrien Mélot
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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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图中最大匹配的平均大小
我们研究了图 G 中最大匹配的平均大小与最大匹配的大小之比 \(\mathcal{I}(G)\)。如果许多最大匹配的大小接近 \(\nu(G)\),那么这个图不变式的值就接近 1。相反,如果许多最大匹配的大小很小,那么(\mathcal {I}(G)\) 接近(\frac{1}{2}\)。我们提出了一种通用技术来确定各种图类的\(\mathcal {I}(G)\) 的渐近行为。为了说明这一技术的使用,我们首先展示了它是如何使我们找到已知的 \(\mathcal {I}(G)\) 的渐近值成为可能的,这些值通常是通过生成函数得到的,然后我们确定了 \(\mathcal {I}(G)\) 对于其他图形族的渐近值,强调了这个图形不变式的可能值在\(\frac{1}{2}\) 和 1 之间的频谱。
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来源期刊
Journal of Combinatorial Optimization
Journal of Combinatorial Optimization 数学-计算机:跨学科应用
CiteScore
2.00
自引率
10.00%
发文量
83
审稿时长
6 months
期刊介绍: The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering. The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.
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