Outer-weakly convex domination number of graphs

Abstract

For a given simple graph $G=((V(G),E(G))$, a set $Ssubseteq V(G)$ is an outer-weakly convex dominating set if every vertex not in $S$ is adjacent to some vertex in $S$ and $V(G)setminus S$ is a weakly convex set. The emph{outer-weakly convex domination number} of a graph $G$, denoted by $widetilde{gamma}_{wcon}(G)$, is the minimum cardinality of an outer-weakly convex dominating set of $G$. An outer-weakly convex dominating set of cardinality $widetilde{gamma}_{wcon}(G)$ will be called a $widetilde{gamma}_{wcon}$-$set$. In this paper, we initiate the study of outer-weakly convex domination as a new variant of graph domination and give some bounds on the outer-weakly convex domination number of a graph. Also, we derived the relations between this parameter to some related domination parameters such as outer-connected domination and outer-convex domination.