Axiomatic characterization of the interval function of a bipartite graph

The axiomatic study on the interval function, induced pathfunction of a connected graph is a well-known area in metric graph theory. In this paper, we present a new axiom: (bp)for anyx,y,z∈V,R(x,y)={x,y}⇒y∈R(x,z)orx∈R(y,z).We study axiom (bp) on the interval function and the induced path function of a connected, simple and finite graph. We present axiomatic characterizations of the interval function of bipartite graphs and complete bipartite graphs. We extend the characterization of the interval function of bipartite graphs to arbitrary bipartite graphs including disconnected bipartite graphs. In addition, we present an axiomatic characterization of the interval function of a forest. Finally, we present an axiomatic characterization of the induced path function of a tree or a 4-cycle using the axiom (bp).