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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 $g(G)$.
The pair $x, y$ openly geodominates $v$ if
$v \neq x, y$ and $v$ is geodominated by $x$, $y$.
A vertex $v$ is link-complete
in $G$ if the subgraph induced by its neighborhood is complete.
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 the open geodomination number
$og(G)$.
We present sharp bounds for $g(G)$ and $og(G)$ in terms
of the number of link-complete vertices in $G$.
It is shown that
for every pair $k$, $n$ of integers
with $4 \leq k \leq n$, there exists a connected graph $G$ of order $n$
without link-complete vertices such that $og(G) = k$.
%It is also shown
%that there is no connected graph $G$ with $g(G) = 2$ and
%$og(G)= 3$.
It is also shown, for every nontrivial connected graph $G$, that
$2 \leq g(G) \leq og(G) \leq 3 g(G)$. Furthermore,
a pair $a$, $b$ of integers with $2 \leq a \leq b \leq 3a$
is realizable as the geodomination and
open geodomination numbers of some nontrivial connected graph
if and only if $(a, b) \neq (2, 3)$.