  For a graph $\Gamma$, a positive integer $s$ and a subgroup $G\leq
\Aut(\Gamma)$, we prove that $G$ is transitive on the set of $s$-arcs of
$\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is
transitive on the set of $(s-1)$-geodesics of its line graph. As applications,
we first prove that the only non-complete locally cyclic $2$-geodesic
transitive graphs are the complete multipartite graph $K_{3[2]}$ and the
icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and
girth 3, and determine which of them are geodesic transitive.
