Γ-induced-paired Dominating Graphs of Paths

Saharath Sanguanpong, Nantapath Trakultraipruk
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Abstract

Let D be a set of vertices of a graph G. Then D is called an induced-paired dominating set if every vertex in G is adjacent to some vertex in D and the subgraph induced by D contains only nonadjacent edges. The upper induced-paired domination number of G, denoted by $\Gamma_{ip}(G)$, is the maximum cardinality of a minimal induced-paired dominating set of G. An induced-paired dominating set of cardinality $\Gamma_{ip}(G)$ is called an $\Gamma_{ip}(G)$-set. We introduce the $\Gamma$-induced-paired dominating graph of G, denoted by $IPD_{\Gamma}$, to be the graph whose vertex set is the set of all $\Gamma_{ip}(G)$-sets, and two $\Gamma_{ip}(G)$-sets are adjacent in $IPD_{\Gamma}(G)$ if one can be obtained from the other by removing one vertex and adding another vertex of G. In this paper, we present the $\Gamma$-induced-paired dominating graphs of all paths.
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Γ-induced-paired支配路径图
设D是图G的一个顶点集合,如果G中的每个顶点都与D中的某个顶点相邻,并且由D诱导的子图只包含非相邻边,则D称为诱导配对支配集。G的上诱导配对支配数,用$\Gamma_{ip}(G)$表示,是G的最小诱导配对支配集的最大基数。基数$\Gamma_{ip}(G)$的诱导配对支配集称为$\Gamma_{ip}(G)$-集。我们引入G的$\Gamma$诱导配对支配图,表示为$IPD_{\Gamma}$,其顶点集是所有$\Gamma_{ip}(G)$-集合的集合,并且如果两个$\Gamma_{ip}(G)$-集合在$IPD_{\Gamma}(G)$中相邻,则其中一个可以通过G的一个顶点减去另一个顶点而得到。本文给出了所有路径的$\Gamma$诱导配对支配图。
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