arXiv 1407.73682014-07-28cs.DMmath.CO

图的临界独立集

Critical Independent Sets of a Graph

Vadim E. Levit, Eugen Mandrescu

备注：: 15 pages; 12 figures. arXiv admin note: substantial text overlap with arXiv:1102.1138

摘要：

设$G$是一个简单图，其顶点集为$V\left( G\right) $ .一个集合$S\subseteq V\left( G\right) $是独立的，如果没有两个顶点从$S$是相邻的，$\mathrm{Ind}(G)$我们的意思是所有$G$的独立集合的家庭。数字$d\left( X\right) =$ $\left\vert X\right\vert -\left\vert N(X)\right\vert $是$X\subseteq V\left( G\right) $的差，如果是$d(A)=\max \{d\left( I\right) :I\in\mathrm{Ind}(G)\}$，则设置$A\in\mathrm{Ind}(G)$至关重要（Zhang，1990）。让我们回想一下以下定义：$\mathrm{core}\left( G\right) $ = $\bigcap$ {S：S是最大独立集}。$\mathrm{corona}\left( G\right)$ = $\bigcup$ {S：S是最大独立集}。$\mathrm{\ker}(G)$ = $\bigcap$ {S：S是临界独立集}。$\mathrm{diadem}(G)$ = $\bigcup$ {S：S是临界独立集}。在本文中，我们提出了各种结构性质的$\mathrm{\ker}(G)$，与$\mathrm{core}\left( G\right) $，$\mathrm{corona}\left( G\right) $，和$\mathrm{diadem}(G)$。

英文摘要：

Let $G$ be a simple graph with vertex set $V\left( G\right) $. A set $S\subseteq V\left( G\right) $ is independent if no two vertices from $S$ are adjacent, and by $\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$. The number $d\left( X\right) =$ $\left\vert X\right\vert -\left\vert N(X)\right\vert $ is the difference of $X\subseteq V\left( G\right) $, and a set $A\in\mathrm{Ind}(G)$ is critical if $d(A)=\max \{d\left( I\right) :I\in\mathrm{Ind}(G)\}$ (Zhang, 1990). Let us recall the following definitions: $\mathrm{core}\left( G\right) $ = $\bigcap$ {S : S is a maximum independent set}. $\mathrm{corona}\left( G\right)$ = $\bigcup$ {S :S is a maximum independent set}. $\mathrm{\ker}(G)$ = $\bigcap$ {S : S is a critical independent set}. $\mathrm{diadem}(G)$ = $\bigcup$ {S : S is a critical independent set}. In this paper we present various structural properties of $\mathrm{\ker}(G)$, in relation with $\mathrm{core}\left( G\right) $, $\mathrm{corona}\left( G\right) $, and $\mathrm{diadem}(G)$.

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。

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英文摘要：

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