### 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.