### k-Geodomination in graphs

For an integer $k \geq 1$,
a vertex $v$ of a graph $G$ is $k$-geodominated by a pair
$x, y$ of vertices in $G$ if $d(x, y) = k$ and
$v$ lies on an $x-y$ geodesic of $G$.
A set $S$ of vertices of $G$ is a $k$-geodominating set if
each vertex $v$ in $V - S$ is $k$-geodominated
by some pair of distinct vertices of $S$.
The minimum cardinality of a $k$-geodominating set of $G$
is its $k$-geodomination number
$g_k(G)$. A vertex $v$ is openly $k$-geodominated by a pair
$x, y$ of distinct vertices in $G$ if $v$ is
$k$-geodominated by $x$ and $y$ and $v \neq x, y$.
A vertex $v$ in $G$ is a $k$-extreme vertex if $v$
is not openly $k$-geodominated by any pair
of vertices in $G$.
A set $S$ of vertices of $G$ is an open $k$-geodominating set of $G$ if
for each vertex $v$ of $G$, either (1) $v$ is $k$-extreme and $v \in S$ or
(2) $v$ is openly $k$-geodominated by some pair of distinct vertices of
$S$.
The minimum cardinality of
an open $k$-geodominating set in $G$
is its open $k$-geodomination number
$og_k(G)$.
It is shown that each triple
$a$, $b$, $k$ of integers with $2 \leq a \leq b$
and $k \geq 2$ is realizable as the
geodomination number and
$k$-geodomination number of some tree.
For each integer $k \geq 1$, we show that
a pair $(a, n)$ of integers is realizable as
the $k$-geodomination number (open $k$-geodomination number)
and order of some nontrivial connected graph
if and only of $2 \leq a = n$ or $2 \leq a \leq n-k+1$.
We investigate how
$k$-geodomination numbers are affected by
adding a vertex.
We show that if $G$ is a nontrivial connected graph of diameter
$d$ with exactly $\ell$
$k$-extreme vertices, then
$\max \ \{2, \ell\} \leq g_k(G) \leq og_k (G)
\leq 3g_k(G) - 2\ell$
for every integer $k$ with $2 \leq k \leq d$.