Finite s-Geodesic Transitive Graphs Which Are Locally Disconnected

Abstract

It was previously shown that there is a bijection between the family of locally disconnected 2-geodesic transitive graphs $$\Gamma $$ and a certain family of partial linear spaces $${\mathcal {S}}(\Gamma )$$ . In this paper, we first determine a relationship between the 2-geodesic transitivity of $$\Gamma $$ and the local s-arc transitivity of the incidence graph of $${\mathcal {S}}(\Gamma )$$ . Next, we give a reduction theorem for the family of locally disconnected s-geodesic transitive graphs.