The induced path function, monotonicity and betweenness

The geodesic interval function II of a connected graph allows an axiomatic characterization involving axioms on the function only, without any reference to distance, as was shown by Nebeský [20]. Surprisingly, Nebeský [23] showed that, if no further restrictions are imposed, the induced path function JJ of a connected graph GG does not allow such an axiomatic characterization. Here J(u,v)J(u,v) consists of the set of vertices lying on the induced paths between uu and vv. This function is a special instance of a transit function. In this paper we address the question what kind of restrictions could be imposed to obtain axiomatic characterizations of JJ. The function JJ satisfies betweenness if w∈J(u,v)w∈J(u,v), with w≠uw≠u, implies u∉J(w,v)u∉J(w,v) and x∈J(u,v)x∈J(u,v) implies J(u,x)⊆J(u,v)J(u,x)⊆J(u,v). It is monotone if x,y∈J(u,v)x,y∈J(u,v) implies J(x,y)⊆J(u,v)J(x,y)⊆J(u,v). In the case where we restrict ourselves to functions JJ that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of JJ by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.