### 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.

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