Geodetic Graphs: Experiments and New Constructions

TL;DR

Geodetic graphs, defined by a unique geodesic between every vertex pair, resist full classification. The authors develop two complementary enumeration strategies—a nauty-driven generation of biconnected graphs and a custom shortest-path-tree backtracking method—to exhaustively enumerate geodetic graphs up to $n\le 25$ vertices and regular geodetic graphs up to $n\le 32$ vertices, discovering two new infinite families. They produce a complete list of biconnected geodetic graphs with at most $25$ vertices, identify two new graphs $F_5$ and $H(2,2,2,0)$, and derive two infinite families from them, alongside rigorous proofs of two new constructions $H(m,n,p,s)$ and $F_k$. The work highlights the scarcity of geodetic graphs, many being subdivisions of smaller graphs, and provides data, methods, and concrete constructions to guide future classifications and growth-rate analyses. These results deepen understanding of geodesic structure in graphs and lay groundwork for exploring subdivisions and new infinite families in larger graphs, with potential implications for related distance-sensitive and fault-tolerant applications.

Abstract

In 1962 Ore initiated the study of geodetic graphs. A graph is called geodetic if the shortest path between every pair of vertices is unique. In the subsequent years a wide range of papers appeared investigating their peculiar properties. Yet, a complete classification of geodetic graphs is out of reach. In this work we present a program enumerating all geodetic graphs of a given size. Using our program, we succeed to find all geodetic graphs with up to 25 vertices and all regular geodetic graphs with up to 32 vertices. This leads to the discovery of two new infinite families of geodetic graphs.

Figures (3)

• Figure 1: Excerpt of the search space. In the first depicted step sibling edges for the three vertices on the bottom right are generated. Note that the two other ways to insert a single sibling edge yield graphs isomorphic to the middle one. In the second step non-sibling neighbours of the vertex on the bottom left are generated. A vertex in the completed state is represented by a filled circle, semi-complete by a half-full circle.
• Figure 2: The family $h(m, n, s)$bosak1978geodetic and the newly discovered family $H(m, n, p, s)$.
• Figure 3: The family $F_k$. The dashed line on the right represents the cut used to obtain any $F_k$ ($k > 5$) from $F_5$ using \ref{['prop:subdivcut']}.

Theorems & Definitions (8)

• lemma thmcounterlemma: Theorem 3.3 in stemple1974geodetic
• lemma thmcounterlemma: Theorem 3.5 in stemple1974geodetic
• lemma thmcounterlemma
• lemma thmcounterlemma: Unique Predecessor Theorem parthasarathy1982some
• lemma thmcounterlemma
• proof
• proposition thmcounterproposition
• proposition thmcounterproposition