The geodesic interval function 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 of a connected graph does not allow such an axiomatic characterization. Here consists of the set of vertices lying on the induced paths between and . 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 . The function satisfies betweenness if , with , implies and implies . It is monotone if implies . In the case where we restrict ourselves to functions that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.