Weakly Convex Domination Subdivision Number of a Graph

Magda Dettlaff, Saeed Kosary, Magdalena Lemańka, Seyed Mahmoud Sheikholeslami

A set $X$ is \emph{weakly convex} in $G$ if for any two vertices $a,b\in X$ there exists an $ab$-geodesic such that all of its vertices belong to $X$. A set $X\subseteq V$ is a \emph{weakly convex dominating set} if $X$ is weakly convex and dominating. The \emph{weakly convex domination number} $\gamma_{\operatorname{wcon}}(G)$ of a graph $G$ equals the minimum cardinality of a weakly convex dominating set in $G$. The \emph{weakly convex domination subdivision number} $\operatorname{sd}_{\gamma_{\operatorname{wcon}}}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and establish upper bounds for it.