### The Forcing Geodetic Number of a Graph

For two vertices $u$ and $v$ of a graph $G$, the set $I(u, v)$ consists
of all vertices lying on some $u-v$ geodesic in $G$. If $S$ is a
set of vertices of $G$, then $I(S)$ is the union of all sets $I(u,v)$ for
$u, v \in S$.
A set $S$ is a geodetic set if $I(S)=V(G)$. A minimum
geodetic set is a geodetic set of minimum
cardinality and this cardinality is the
geodetic number $g(G)$.
A subset $T$ of a minimum geodetic set $S$ is called a forcing subset
for $S$ if $S$ is the unique minimum geodetic set containing $T$.
The forcing geodetic number $f_{G}(S)$ of $S$ is the minimum
cardinality among the
forcing subsets of $S$, and the forcing geodetic number
$f(G)$ of $G$ is the minimum forcing geodetic number
among all minimum geodetic sets of $G$. The forcing geodetic numbers
of several classes of graphs are determined.
For every graph $G$, $f(G) \leq g(G)$.
It is shown that for all integers $a, b$ with $0\leq a\leq b$, a
connected graph $G$ such that $f(G)=a$ and $g(G)=b$ exists if and only
if $(a,b)\notin\{(1,1),(2,2)\}$.