The axiomatic approach with the interval function and induced path transit function of a connected graph is an interesting topic in metric and related graph theory. In this paper, we introduce a new axiom: (bp) for any x, y, z ∈ V, R(x, y) = {x, y} ⇒ y ∈ R(x, z) or x ∈ R(y, z). We study axiom (bp) on the interval function and the induced path transit function of a connected, simple and finite graph. We present axiomatic characterizations of the interval function of bipartite graphs and complete bipartite graphs. Further, we present an axiomatic characterization of the induced path transit function of a tree or a 4-cycle. © Springer International Publishing AG 2017.