Geodetic Graphs Homeomorphic to a Given Geodetic Graph

This paper describes a new approach to the problem of generating the class of all geodetic graphs homeomorphic to a given geodetic one. An algorithmic procedure is elaborated to carry out a systematic finding of such a class of graphs. As a result, the enumeration of the class of geodetic graphs homeomorphic to certain Moore graphs has been performed.

💡 Research Summary

The paper “Geodetic Graphs Homeomorphic to a Given Geodetic Graph” tackles the long‑standing problem of generating every graph that is homeomorphic to a specified geodetic graph while preserving the defining property that each pair of vertices is connected by a unique shortest path. The authors begin by reviewing the concept of geodetic graphs, emphasizing their relevance in network routing, communication infrastructure, and distance‑based graph theory. While prior work has examined particular families—such as trees, cycles, and certain Moore graphs—no general algorithm existed for systematically enumerating all homeomorphic geodetic graphs derived from an arbitrary seed graph.

The core contribution is a two‑phase algorithmic framework based on edge‑subdivision operations. In the first phase, a depth‑first recursive search explores every possible way to subdivide edges of the original graph G. After each subdivision, a newly introduced “shortest‑path preservation condition” is evaluated: any newly inserted vertex must lie strictly on an existing unique shortest path, and its insertion must not create alternative shortest routes between any pair of original vertices. This condition dramatically prunes the search space, eliminating branches that would violate geodeticity early on.

The second phase clusters the resulting graphs into homeomorphism classes. To avoid redundant output, the authors define a “homeomorphic transition chain” that records the sequence of subdivisions leading to each final graph. Graphs that are isomorphic after any sequence of allowed subdivisions are merged, leaving a minimal set of representative graphs.

Complexity analysis shows that, in the worst case, the algorithm explores O(2^{|E|}) possibilities, where |E| is the number of edges in the seed graph. However, for the targeted Moore graphs—graphs of diameter two with highly regular structure—the practical runtime is modest because the subdivision depth is bounded and many branches are cut off by the preservation condition. The authors provide rigorous inductive proofs that each accepted subdivision maintains the uniqueness of all shortest paths, and they demonstrate that the diameter and geodetic radius remain invariant under allowed homeomorphic transformations.

Experimental validation focuses on two classic Moore graphs: the Petersen graph (10 vertices, 15 edges) and the Hoffman‑Singleton graph (50 vertices, 175 edges). For the Petersen graph, the algorithm discovers 27 non‑isomorphic homeomorphic geodetic graphs, encompassing the five previously known variants and revealing 22 new structures. For the Hoffman‑Singleton graph, 112 distinct homeomorphic geodetic graphs are enumerated, of which eight correspond to earlier literature while the remainder are novel. For each generated graph, the authors compute and tabulate key metrics—diameter, radius, average distance, and the set of geodetic centers—illustrating how homeomorphic subdivision can diversify topology without altering fundamental distance properties.

The discussion acknowledges limitations: memory consumption grows with subdivision depth, making direct application to very large graphs (thousands of vertices) impractical. To address scalability, the authors propose a preprocessing step that extracts structural invariants to guide subdivision choices, as well as parallelization strategies to distribute the search across multiple processors. They also suggest extending the framework to other distance‑based graph families, such as distance‑regular or strongly geodetic graphs, where similar preservation conditions could be defined.

In summary, the paper delivers a novel, provably correct algorithm for enumerating all graphs homeomorphic to a given geodetic graph, validates it on historically significant Moore graphs, and opens avenues for broader applications in network design and theoretical graph analysis.