Lobachevskii Journal of Mathematics Pub Date : 2021-06-04 , DOI: 10.1134/s1995080221040156 F. Movahedi , M. H. Akhbari , S. Alikhani
Abstract
Let \(G=(V,E)\) be a simple graph. A set \(D\subseteq V\) is a \(2\)-dominating set of \(G\), if every vertex of \(V\setminus D\) has at least two neighbors in \(D\). The \(2\)-domination number of a graph \(G\), is denoted by \(\gamma_{2}(G)\) and is the minimum size of the \(2\)-dominating sets of \(G\). In this paper, we count the number of \(2\)-dominating sets of \(G\). To do this, we consider a polynomial which is the generating function for the number of \(2\)-dominating sets of \(G\) and call it \(2\)-domination polynomial. We study some properties of this polynomial. Furthermore, we compute the \(2\)-domination polynomial for some of the graph families.
中文翻译:
图的 2-支配集数和 2-支配多项式
摘要
令\(G=(V,E)\)是一个简单的图。如果\(V\setminus D\) 的每个顶点在\(D\)中至少有两个邻居,则集合\(D\subseteq V\)是\(2\)支配集\(G \) . 的\(2 \) -domination的曲线图的数\(G \) ,是由表示为\(\ gamma_ {2}(G)\)并且是最小尺寸\(2 \) -dominating套\ (G\)。在本文中,我们计算\(2\)支配\(G\)集的数量。为此,我们考虑一个多项式,它是\(2\)-支配\(G\) 的集合并将其称为\(2\) -支配多项式。我们研究这个多项式的一些性质。此外,我们计算了一些图族的\(2\)支配多项式。