Weakly convex domination subdivision number of a graph

Abstract

A set $X$ is {\em 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 {\em weakly
convex dominating set} if $X$ is weakly convex and dominating. The
{\em weakly convex domination number} $\gamma_{\rm wcon}(G)$ of a
graph $G$ equals the minimum cardinality of a weakly convex
dominating set in $G$. {\em The weakly convex domination
subdivision number} sd$_{\gamma_{\rm wcon}}(G)$ 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 we establish upper bounds
for it.