Associative triples and homomorphisms between travel groupoids on finite geodetic graphs

The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set V and a binary operation * on V satisfying two axioms. We can associate a graph with a travel groupoid. We say that a graph G has a travel groupoid if the graph associated with the travel groupoid is equal to G. Nebeský gave a characterization of finite graphs having a travel groupoid.

The notion of travel groupoids was introduced by Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set V and a binary operation $$*$$ on V satisfying two axioms. For a travel groupoid, we can associate a graph. We say that a graph G has a travel groupoid if the graph associated with the travel groupoid is equal to G. Nebeský gave a characterization for finite graphs to have a travel groupoid. In this paper, we introduce the notion of T-neighbor systems on a graph and give a characterization of travel groupoids on a graph in terms of T-neighbor systems.