A geodetic graph is a connected graph with a unique shortest path between every pair of vertices. Ore (1962) introduced the term and asked for a characterization extending the familiar property of trees.
The numbers of geodetic graphs on 
, 2, ... vertices are 1,
1, 2, 4, 10, 23, 66, 185, 586, 1880, 6360, 21975, 78230, 283087, 1043329, ... (OEIS
A337179).
Examples of geodetic graphs include barbell graphs, block graphs, cacti graph
lacking even cycles (Gorovoy and Zmiaikou 2021), complete
graphs 
(Gorovoy and Zmiaikou 2021), odd cycle graphs 
(Gorovoy and Zmiaikou 2021),
lollipop graphs, trees
(Gorovoy and Zmiaikou 2021), triangular snake
graphs, and windmill graphs.
Stober and Weiß (2026) enumerate all biconnected graphs with the geodetic property through 25 vertices,
obtaining 149 in total. Their enumeration also gives connected
counts through 22 vertices and all regular
examples through 32 vertices. Their Stober-Weißgraphs
form a new infinite family 
obtained from three constituent geodetic pieces by
identifying specified cliques. The smallest member of
this family, 
,
has 24 vertices.
-Geodetic Graphs and Their Application to the Topological Design
of Computer Networks." In Proc. Argentinian Workshop on Theoretical Computer
Science, 28 JAIIO-WAIT'99. pp. 187-203, 1999.Gorovoy, D. and
Zmiaikou, D. "On Graphs with Unique Geoodesics and Antipodes." 19 Nov 2021.