Finite $s$-geodesic-transitive digraphs

This paper initiates the investigation of the family of $(G,s)$-geodesic-transitive digraphs with $s\geq 2$. We first give a global analysis by providing a reduction result. Let $\Gamma$ be such a digraph and let $N$ be a normal subgroup of $G$ maximal with respect to having at least $3$ orbits. Then the quotient digraph $\Gamma_N$ is $(G/N,s')$-geodesic-transitive where $s'=\min\{s,\diam(\Gamma_N)\}$, $G/N$ is either quasiprimitive or bi-quasiprimitive on $V(\Gamma_N)$, and $\Gamma_N$ is either directed or an undirected complete graph. Moreover, it is further shown that if $\Gamma$ is not $(G,2)$-arc-transitive, then $G/N$ is quasiprimitive on $V(\Gamma_N)$. On the other hand, we also consider the case that the normal subgroup $N$ of $G$ has one orbit on the vertex set. We show that if $N$ is regular on $V(\Gamma)$, then $\Gamma$ is a circuit, and particularly each $(G,s)$-geodesic-transitive normal Cayley digraph with $s\geq 2$, is a circuit. Finally, we investigate $(G,2)$-geodesic-transitive digraphs with either valency at most 5 or diameter at most 2. Let $\Gamma$ be a $(G,2)$-geodesic-transitive digraph. It is proved that: if $\Gamma$ has valency at most $5$, then $\Gamma$ is $(G,2)$-arc-transitive; if $\Gamma$ has diameter $2$, then $\Gamma$ is a balanced incomplete block design with the Hadamard parameters.