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 $⟨D{⟩}_G$ 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.