On the outer-connected domination in graphs

Abstract

A set S of vertices of a graph G is an outer-connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by V∖S is connected. The outer-connected domination number $\widetilde{\gamma}_{c}(G)$ is the minimum size of such a set. We prove that if δ(G)≥2 and diam (G)≤2, then $\widetilde{\gamma}_{c}(G)\le (n+1)/2$, and we study the behavior of $\widetilde{\gamma}_{c}(G)$ under an edge addition.