The Domination Parameters of The Corona and Its Generalization

Abstract

Let \(D\) be a dominating set of a simple graph \(G = (V, E)\). If the subgraph \((V – D)_G\)induced by the set \(V – D\) is disconnected, then \(D\) is called a split dominating set of \(G\), and if \(\langle D\rangle_G\) has no edges, then \(D\) is an independent dominating set of \(G\). If every vertex in \(V\) is adjacent to some vertex of \(D\) in \(G\), then \(D\) is a total dominating set of \(G\). The split domination number \(\gamma_s(G)\), independent domination number \(i(G)\), and total domination number \(\gamma_t(G)\) equal the minimum cardinalities of a split, independent, and total dominating set of \(G\), respectively. The concept of split domination was first defined by Kulli and Janakiram in 1997 [4], while total domination was introduced by Cockayne, Dawes, and Hedetniemi in 1980 [2].

In this paper, we study the split, independent, and total domination numbers of corona \(G \circ H\) and generalized coronas \(kG \circ H\) of graphs.