Geodomination in graphs (Q2706953)

A subset \(S\) of the vertex set \(V(G)\) of a graph \(G\) is called geodominating in \(G\), if for each \(v\in V(G)\) there exist vertices \(x\), \(y\) in \(S\) such that either \(v\in \{x,y\}\), or \(v\) lies on a \(x\)-\(y\) geodesic in \(G\). A vertex \(v\) is link-complete in \(G\), if its neighbourhood induces a complete subgraph of \(G\). A set \(S\subseteq V(G)\) is called open geodominating, if for each \(v\in V(G)\) either \(v\) is link-complete and \(v\in S\), or there exist vertices \(x\), \(y\) in \(S\) such that \(v\not\in\{x, y\}\) and \(v\) lies on a \(x\)-\(y\) geodesic in \(G\). The minimum number of vertices of a geodominating (or open geodominating) set in \(G\) is called the geodomination number \(g(G)\) (or the open geodomination number \(og(G)\)) of \(G\). The chain of inequalities \(\max\{2, k\}\leq g(G)\leq og(G)\leq 3g(G)- 2k\) is presented, where \(k\) is the number of link-complete vertices of \(G\). If \(a\), \(b\) are integers, \(2\leq a\leq b\leq 3a\), then a nontrivial connected graph \(G\) with \(g(G)= a\) and \(og(G)= b\) exists if and only if \((a,b)\neq (2,3)\).