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The Upper Forcing Geodetic Number of a Graph

For vertices $u$ and $v$ in a nontrivial
connected graph $G$,
the closed interval $I[u, v]$ consists of
$u$, $v$, and all vertices lying in some $u-v$ geodesic of $G$. For
$S \sbe V(G)$, the set $I[S]$ is the union of all
sets $I[u, v]$ for $u, v \in S$. A set $S$ of vertices
of a graph $G$ is a
geodetic set in $G$ if
$I[S]=V(G)$. The minimum cardinality
of a geodetic set in $G$ is its geodetic number $g(G)$.
A subset $T$ of a minimum geodetic set $S$ in a
graph $G$ is a forcing subset
for $S$ if $S$ is the unique minimum geodetic set containing $T$.
The forcing geodetic number $f(S)$ of $S$ in $G$ is the minimum
cardinality of a forcing subset for $S$, and the
upper forcing geodetic number
$f^+(G)$ of the graph $G$ is the maximum forcing geodetic number
among all minimum geodetic sets of $G$. Thus $0 \leq f^+(G) \leq g(G)$
for every graph $G$.
The upper forcing geodetic numbers of several classes of graphs
are determined.
It is shown that for every pair $a, b$ of integers with
$0 \leq a \leq b$ and $b \geq 1$,
there exists a connected graph $G$ with
$f^+(G) = a$ and $g(G) = b$ if and only if
$(a, b) \notin \{(1, 1), (2, 2)\}$.