The induced path function, monotonicity and betweenness

The induced path function J(u, v) of a graph consists of the set of all vertices lying on the induced paths between vertices u and v. This function is a special instance of a transit function. The function J satisfies betweenness if w ∈ J(u, v) implies u / ∈ J(w, v) and x ∈ J(u, v) implies J(u, x) ⊆ J(u, v), and it is monotone if x, y ∈ J(u, v) implies J(x, y) ⊆ J(u, v). The induced path function of a connected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.