Allagan, Julian A. D.; Bobga, Benkam Dominion of some graphs. (English) Zbl 1469.05076 Int. J. Math. Comput. Sci. 16, No. 4, 1709-1720 (2021). Summary: Given a graph \(G=(V,E)\), a subset \(S\subseteq V\) is a dominating set if every vertex in \(V\setminus S\) is adjacent to some vertex in \(S\). The dominating set with the least cardinality, \(\gamma\), is called a \(\gamma\)-set which is commonly known as a minimum dominating set. The dominion of a graph \(G\), denoted by \(\zeta(G)\), is the number of its \(\gamma\)-sets. Some relations between these two seemingly distinct parameters are established. In particular, we present the dominions of paths, some cycles and the join of any two graphs. MSC: 05C30 Enumeration in graph theory 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) Keywords:domination; dominion PDFBibTeX XMLCite \textit{J. A. D. Allagan} and \textit{B. Bobga}, Int. J. Math. Comput. Sci. 16, No. 4, 1709--1720 (2021; Zbl 1469.05076) Full Text: Link