Global outer connected domination number of a graph

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.