A forbidden subgraph characterization of some graph classes using betweenness axioms

Let I G ( x , y ) and J G ( x , y ) be the geodesic and induced path intervals 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 V : (i) ( u , y ) , y ∈ R ( x , v ) , x ≠ y , | R ( u , v ) | > 2 ⇒ x ∈ R ( u , v ) ; ( u , v ) ⇒ R ( u , x ) ∩ R ( x , v ) = { x } ; ( u , y ) , y ∈ R ( x , v ) , x ≠ y ⇒ x ∈ R ( u , v ) . aracterize the class of graphs for which I G satisfies (i), the class for which J G satisfies (ii) and the class for which I G or J G satisfies (iii). The characterization is given in terms of forbidden induced subgraphs. It turns out that the class of graphs for which I G satisfies (i) is a proper subclass of distance hereditary graphs and the class for which J G satisfies (ii) is a proper superclass of distance hereditary graphs. We also give an axiomatic characterization of chordal and Ptolemaic graphs.