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

The subdivision graph $S(\mathrm{\Sigma })$ of a connected graph $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$ such that $S(\mathrm{\Sigma })$ is locally $(G,s)$-distance transitive for $s\leq 2, diam(\Sigma)-1$andsome$G\leq Aut(\Sigma)$. In this paper, we solve the remaining cases by classifying all the graphs $\mathrm{\Sigma }$ such that the subdivision graphs is locally $(G,s)$-distance transitive for $s\geq 2, diam(\Sigma)$andsome$G\leq Aut(\Sigma)$. In particular, their subdivision graph are always locally $G$-distance transitive, except for the complete graphs.