arXiv 1411.78182014-11-28math.CO

关于图的完美控制与拟完美控制

On Perfect and Quasiperfect Domination in Graphs

José Cáceres, Carmen Hernando, Mercè Mora, Ignacio M. Pelayo, María Luz Puertas

备注：: 14 pages, 9 figures

摘要：

一个图$G=(V,E)$中的子集$S\subseteq V$是$k$ -拟完美控制集（对于$k\geq 1$），如果每个不在$S$中的顶点至少与$S$中的一个且至多与$k$顶点相邻.用$γ_ {\stackrel{}{1k}}(G)$表示$G$中最小$k$ -拟完美控制集的基数.这些集合最早由Chellali et al.（2013）引入，作为完美控制概念的推广，并允许我们构造一个准完美控制数的递减链$ n \ge γ_ {\stackrel{}{11}}(G) \ge γ_ {\stackrel{}{12}}(G)\ge \ldots \ge γ_ {\stackrel{}{1Δ}}(G)=γ(G)$，以表明$G$离完美控制有多远。本文研究短链图即$γ_ {\stackrel{}{12}}(G)=γ(G)$的性质、存在性和实现。在这些图中，我们可以找到上图、无爪图和具有$Δ(G)$极值的图。

英文摘要：

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $γ_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge γ_ {\stackrel{}{11}}(G) \ge γ_ {\stackrel{}{12}}(G)\ge \ldots \ge γ_ {\stackrel{}{1Δ}}(G)=γ(G)$ in order to indicate how far is $G$ from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, $γ_ {\stackrel{}{12}}(G)=γ(G)$. Among them, one can find cographs, claw-free graphs and graphs with extremal values of $Δ(G)$.

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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