Extreme Geodesic Graphs

For two vertices $u$ and $v$ of a graph $G$, the {\it closed interval} $I[u, v]$ consists of $u$, $v$, and all vertices lying in some $u-v$ geodesic of $G$, while 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 $G$ for which $I[S]=V(G)$ is a geodetic set for $G$, and the minimum cardinality of a geodetic set is the geodetic number $g(G)$. A vertex $v$ in $G$ is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in $G$ is its extreme order $ex(G)$. A graph $G$ is an extreme geodesic graph if $g(G) = ex(G)$, that is, if every vertex lies on a $u-v$ geodesic for some pair $u, v$ of extreme vertices. It is shown that every pair $a, b$ of integers with $0 \leq a \leq b$ is realizable as the extreme order and geodetic number, respectively, of some graph. For positive integers $r, d,$ and $k \geq 2$, it is shown that there exists an extreme geodesic graph $G$ of radius $r$, diameter $d$, and geodetic number $k$. Also, for integers $n, d, $ and $k$ with $2 \leq d < n$, $2 \leq k < n$, and $n -d - k +1 \geq 0$, there exists a connected extreme geodesic graph $G$ of order $n$, diameter $d$, and geodetic number $k$. We show that every graph of order $n$ with geodetic number $n-1$ is an extreme geodesic graph. On the other hand, for every pair $k, n$ of integers with $2 \leq k \leq n-2$, there exists a connected graph of order $n$ with geodetic number $k$ that is not an extreme geodesic graph.