On Geodomination in Graphs

A pair $x, y$ of vertices in a nontrivial connected graph $G$ is said to geodominate a vertex $v$ of $G$ if either $v \in \{x, y\}$ or $v$ lies in an $x-y$ geodesic of $G$. A set $S$ of vertices of $G$ is a geodominating set if every vertex of $G$ is geodominated by some pair of vertices of $S$. The cardinality of a minimum geodominating set in $G$ is the geodomination number $g(G)$. A vertex of $G$ is link-complete if the subgraph induced by its neighborhood is complete. A pair $x, y$ of vertices in $G$ is said to openly geodominate a vertex $v$ of $G$ if $v \neq x, y$ and $v$ is geodominated by $x$ and $y$. A set $S$ is an open geodominating set of $G$ if for each vertex $v$, either (1) $v$ is link-complete and $v \in S$ or (2) $v$ is openly geodominated by some pair of vertices of $S$. The cardinality of a minimum open geodominating set in $G$ is its open geodomination number $og(G)$. We study the minimum geodominating sets in a connected graph and how the geodomination number of a graph is affected by adding a vertex to the graph. It is shown that the geodomination number $g(G \times K_s)$ of the Cartesian product of a connected graph $G$ and $K_s$ is bounded below by $\max \{s, g(G)\}$ for all integers $s \geq 1$ and this bound is sharp. Also, the open geodomination numbers of several classes of graphs are determined.