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.
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Funding
M. H. Akhbari is partially supported by the Program of development of Scientific and Educational Mathematical Center of Volga Region (project no. 075-02-2020-1478).
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(Submitted by M. M. Arslanov)
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Movahedi, F., Akhbari, M.H. & Alikhani, S. The Number of 2-dominating Sets, and 2-domination Polynomial of a Graph. Lobachevskii J Math 42, 751–759 (2021). https://doi.org/10.1134/S1995080221040156
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DOI: https://doi.org/10.1134/S1995080221040156