论文标题
关于统治的优秀图：由最小统治集引起的子图
Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets
论文作者
论文摘要
图$ g =（v，e）$是$γ$ -EXCELLENT，如果$ v $是$ g $的所有$γ$集合的结合，其中$γ$代表统治号码。令$ \ mathcal {i} $为所有相互非态图的集合，$ \ emptyset \ not = \ natercal {h} \ subsetneq \ subsetneq \ mathcal {i} $。在本文中，我们启动了$ \ MATHCAL {H} $ - $γ$ -Excellent图的研究，我们将其定义如下。图$ g $是$ \ mathcal {h} $ - $γ$ - excellent如果保持以下内容：（i）对于\ nathcal {h} $中的每一个$ h \，对于每一个$ x \ in v（g）$中的每个$ x \ in v（g）$中都有一个诱导的$ h_x $ h_x $ h_x $ g $ g $ h_x $ h_x $ and $ h_x $和is is is is is is is is is sOMORPHIC， $ v（h_x）$是$ g $的一些$γ$集的子集，（ii）$ g $ $ g $的每个引起的子图$ h $的顶点集，这对于$ \ nathcal {h} $的某些元素是同构的，是一些$γ$ $ g $的子集。对于每个众所周知的图表，包括周期，树木和两个图形的笛卡尔产品，我们描述了其最大的$ \ Mathcal {h} \ subsetNeq \ Mathcal {i} $，该图为$ \ MATHCAL {H} $ - $umγ$ -Excellent。提出了$γ$ - excellent常规图的结果，并提出了图形的广义词素图。提出了几个开放问题和问题。
A graph $G=(V,E)$ is $γ$-excellent if $V$ is a union of all $γ$-sets of $G$, where $γ$ stands for the domination number. Let $\mathcal{I}$ be a set of all mutually nonisomorphic graphs and $\emptyset \not= \mathcal{H} \subsetneq \mathcal{I}$. In this paper we initiate the study of the $\mathcal{H}$-$γ$-excellent graphs, which we define as follows. A graph $G$ is $\mathcal{H}$-$γ$-excellent if the following hold: (i) for every $H \in \mathcal{H}$ and for each $x \in V(G)$ there exists an induced subgraph $H_x$ of $G$ such that $H$ and $H_x$ are isomorphic, $x \in V(H_x)$ and $V(H_x)$ is a subset of some $γ$-set of $G$, and (ii) the vertex set of every induced subgraph $H$ of $G$, which is isomorphic to some element of $\mathcal{H}$, is a subset of some $γ$-set of $G$. For each of some well known graphs, including cycles, trees and some cartesian products of two graphs, we describe its largest set $\mathcal{H} \subsetneq \mathcal{I}$ for which the graph is $\mathcal{H}$-$γ$-excellent. Results on $γ$-excellent regular graphs and a generalized lexicographic product of graphs are presented. Several open problems and questions are posed.
