围长至少为4的非常好覆盖图和局部最大稳定集拟阵
Very Well-Covered Graphs of Girth at least Four and Local Maximum Stable Set Greedoids
摘要：
图$G$中的\textit{maximum stable set}是具有最大基数的稳定集。$S$是$G$的一个\textit{local maximum stable set}，如果$S$是由$S\cup N(S)$诱导的子图的最大稳定集，则我们写为$S\inΨ(G)$，其中$N(S)$是$S$的邻居。内姆豪泽和特罗特（1975），证明了任何$S\inΨ(G)$都是$G$的最大稳定集的子集。在（Levit & Mandrescu，2002）中，我们证明了森林$T$的家族$Ψ(T)$在其顶点集上形成一个广义拟阵。$G$是二分的、无三角形的、良好覆盖的，而$Ψ(G)$是greedoid的情况分别在（Levit & Mandrescu，2002）、（Levit & Mandrescu，2004）、（Levit & Mandrescu，2007）中进行了分析。本文证明了：如果$G$是围长为$\geq4$的覆盖很好的图，则族$Ψ(G)$是广义拟阵当且仅当$G$有唯一的完美匹配.
英文摘要：
A \textit{maximum stable set} in a graph $G$ is a stable set of maximum cardinality. $S$ is a \textit{local maximum stable set} of $G$, and we write $S\inΨ(G)$, if $S$ is a maximum stable set of the subgraph induced by $S\cup N(S)$, where $N(S)$ is the neighborhood of $S$. Nemhauser and Trotter Jr. (1975), proved that any $S\inΨ(G)$ is a subset of a maximum stable set of $G$. In (Levit & Mandrescu, 2002) we have shown that the family $Ψ(T)$ of a forest $T$ forms a greedoid on its vertex set. The cases where $G$ is bipartite, triangle-free, well-covered, while $Ψ(G)$ is a greedoid, were analyzed in (Levit & Mandrescu, 2002),(Levit & Mandrescu, 2004),(Levit & Mandrescu, 2007), respectively. In this paper we demonstrate that if $G$ is a very well-covered graph of girth $\geq4$, then the family $Ψ(G)$ is a greedoid if and only if $G$ has a unique perfect matching.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
