The convex domination subdivision number of a graph

Abstract

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belong to $X$ for any twovertices $a,bin X$. A set $X$ is a convex dominating set if it isconvex and dominating set. The {em convex domination number}$gamma_{rm con}(G)$ of a graph $G$ equals the minimumcardinality of a convex dominating set in $G$. {em The convexdomination subdivision number} sd$_{gamma_{rm con}}(G)$ is theminimum number of edges that must be subdivided (each edge in $G$can be subdivided at most once) in order to increase the convexdomination number. In this paper we initiate the study of convexdomination subdivision number and we establish upper bounds forit.