Global outer connected domination number of a graph

Abstract

For a given graph \(G=(V,E)\), a dominating set \(D \subseteq V(G)\) is said to be an outer connected dominating set if \(D=V(G)\) or \(G-D\) is connected. The outer connected domination number of a graph \(G\), denoted by \(\widetilde{\gamma}_c(G)\), is the cardinality of a minimum outer connected dominating set of \(G\). A set \(S \subseteq V(G)\) is said to be a global outer connected dominating set of a graph \(G\) if \(S\) is an outer connected dominating set of \(G\) and \(\overline G\). The global outer connected domination number of a graph \(G\), denoted by \(\widetilde{\gamma}_{gc}(G)\), is the cardinality of a minimum global outer connected dominating set of \(G\). In this paper we obtain some bounds for outer connected domination numbers and global outer connected domination numbers of graphs. In particular, we show that for connected graph \(G\ne K_1\),  \( \max\{{n-\frac{m+1}{2}}, \frac{5n+2m-n^2-2}{4}\} \leq \widetilde{\gamma}_{gc}(G) \leq \min\{m(G),m(\overline G)\}\). Finally, under the conditions, we show the equality  of global outer connected domination numbers and outer connected domination numbers for family of trees.