T-partition systems and travel groupoids on a graph

The notion of travel groupoids was introduced by L. Nebesk\'y 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 $G$. Nebesk\'y gave a characterization for finite graphs to have a travel groupoid. In this paper, we introduce the notion of T-partition systems on a graph and give a characterization of travel groupoids on a graph in terms of T-partition systems.