Triangle path transit functions, betweenness and pseudo-modular graphs

The geodesic and induced path transit functions are the two well-studied interval functions in graphs. Two important transit functions related to the geodesic and induced path functions are the triangle path transit functions which consist of all vertices on all u,v-shortest (induced) paths or all vertices adjacent to two adjacent vertices on all u,v-shortest (induced) paths, for any two vertices u and v in a connected graph G. In this paper we study the two triangle path transit functions, namely the IΔ and JΔ on G. We discuss the betweenness axioms, for both triangle path transit functions. Also we present a characterization of pseudo-modular graphs using the transit function IΔ by forbidden subgraphs.