A forbidden subgraph characterization of some graph classes using betweenness axioms

Let IG(x, y) and JG(x, y) be the geodesic and induced path interval between x and y in a connected graph G, respectively. The following three betweenness axioms are considered for a set V and R : V × V → 2 (i) x ∈ R(u, y), y ∈ R(x, v), x 6= y, |R(u, v)| > 2 ⇒ x ∈ R(u, v) (ii) x ∈ R(u, v) ⇒ R(u, x) ∩R(x, v) = {x} (iii)x ∈ R(u, y), y ∈ R(x, v), x 6= y,⇒ x ∈ R(u, v). We characterize the class of graphs for which IG satisfies (i), and the class for which JG satisfies (ii) and the class for which IG or JG satisfies (iii). The characterization is given by forbidden induced subgraphs. The class of graphs that we characterize include a proper subclass and superclass of distance hereditary graphs and we give an axiomatic characterization of chordal and Ptolemaic graphs.