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

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