The axiomatic approach with the interval function, induced path transit function and all-paths transit function of a connected graph form a well studied area in metric and related graph theory. In this paper we introduce the first order axiom: (cp) For any pairwise distinct vertices a, b, c, d ∈ V b ∈ R(a, c) and b ∈ R(a, d) ⇒ c ∈ R(b, d) or d ∈ R(b, c). We study this new axiom on the interval function, induced path transit function and all-paths transit function of a connected simple and finite graph. We present characterizations of claw and paw-free graphs using this axiom on standard path transit functions on graphs, namely the interval function, induced path transit function and the all-paths transit function. The family of 2-connected graphs for which the axiom (cp) is satisfied on the interval function and the induced path transit function are Hamiltonian. Additionally, we study arbitrary transit functions whose underlying graphs are Hamiltonian. © Springer International Publishing Switzerland 2016.