THE FORCING CONVEX DOMINATION NUMBER OF A GRAPH

Let \textit{G} be a connected graph and \textit{D} a minimum convex domination set of $G$. A subset $T \subseteq D$ is called a forcing subset of \textit{D}, if \textit{D} is the unique minimum convex dominating set containing \textit{T}. A forcing subset for \textit{D} of minimum cardinality is a minimum forcing subset of \textit{D}. The forcing convex domination number of \textit{D}, denoted by $\gamma_{con}(D)$, is the cardinality of a minimum forcing subset of \textit{D}. The forcing convex domination number of \textit{G}, denoted by $f_{{\gamma}con}(G)$ and is defined by $f_{\gamma con}(G)$ = min $\lbrace f_{\gamma con}(D) \rbrace$, where the minimum is taken over all minimum convex dominating sets \textit{D} in \textit{G}. Some general properties satisfied by this concepts are studied. The forcing fair dominating number of certain standard graphs are determined. It is shown that for every pair $a,b$ of integers with $0 \leq a < b$, there exists a connected graph \textit{G} such that $f_{{\gamma}con}(G) = a$ and ${\gamma}_{con}(G) = b$.