A note on 3-Steiner intervals and betweenness

The geodesic and geodesic interval, namely the set of all vertices lying on geodesics between a pair of vertices in a connected graph, is a part of folklore in metric graph theory. It is also known that Steiner tree of a (multi) set with k (k > 2) vertices, generalizes geodesics. In [1] the authors studied the k-Steiner intervals S(u1, u2, . . . , uk) on connected graphs (k ≥ 3) as the k-ary generalization of the geodesic intervals. The analogous betweenness axiom (b2) and the monotone axiom (m) were generalized from binary to k-ary functions as: for any u1, . . . , uk, x, x1, . . . , xk ∈ V (G) which are not necessarily distinct, (b2) x ∈ S(u1, u2, . . . , uk) ⇒ S(x, u2, . . . , uk) ⊆ S(u1, u2, . . . , uk), (m) x1, . . . , xk ∈ S(u1, . . . , uk) ⇒ S(x1, . . . , xk) ⊆ S(u1, . . . , uk). The authors conjectured in [1] that the 3-Steiner interval on a connected graph G satisfies the betweenness axiom (b2) if and only if each block of G is geodetic of diameter at most 2. In this paper we settle this conjecture. For this we show that there exists an isometric cycle of length 2k + 1, k > 2, in every geodetic block of diameter at least 3. We also introduce another axiom (b2(2)), which is meaningful only to 3-Steiner intervals and show that this axiom is equivalent to the monotone axiom.