A classification of graphs whose subdivision graphs are locally $G$-distance transitive

TL;DR

This paper classifies graphs whose subdivision graphs are locally G-distance transitive for cases where the parameter s is at least twice the diameter, completing previous partial classifications.

Contribution

It provides a complete classification of graphs with subdivision graphs that are locally G-distance transitive for all relevant s values.

Findings

Subdivision graphs are locally G-distance transitive except for complete graphs.

Complete graphs are the only exceptions in the classification.

The classification completes previous work by covering remaining cases.

Abstract

The subdivision graph $S(\Sigma)$ of a connected graph $\Sigma$ is constructed by adding a vertex in the middle of each edge. In a previous paper written with Cheryl E. Praeger, we characterised the graphs $\Sigma$ such that $S(\Sigma)$ is locally $(G,s)$-distance transitive for $s\leq 2\, diam(\Sigma)-1$ and some $G\leq Aut(\Sigma)$. In this paper, we solve the remaining cases by classifying all the graphs $\Sigma$ such that the subdivision graphs is locally $(G,s)$-distance transitive for $s\geq 2\, diam(\Sigma)$ and some $G\leq Aut(\Sigma)$. In particular, their subdivision graph are always locally $G$-distance transitive, except for the complete graphs.