For a graph Ξ, a positive integer s and a subgroup G\leq
\Aut(\Gamma), we prove that G is transitive on the set of s-arcs of
Ξ if and only if Ξ has girth at least 2(sβ1) and G is
transitive on the set of (sβ1)-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 K3[2]β and the
icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and
girth 3, and determine which of them are geodesic transitive