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 trees of a (multi) set with k ( k > 2 ) vertices, generalize geodesics. In Brešar et al. (2009) [1], the authors studied the k -Steiner intervals S ( u 1 , u 2 , … , u k ) 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 follows. For any u 1 , … , u k , x , x 1 , … , x k ∈ V ( G ) which are not necessarily distinct, (b2) x ∈ S ( u 1 , u 2 , … , u k ) ⇒ S ( x , u 2 , … , u k ) ⊆ S ( u 1 , u 2 , … , u k ) , (m) x 1 , … , x k ∈ S ( u 1 , … , u k ) ⇒ S ( x 1 , … , x k ) ⊆ S ( u 1 , … , u k ) . thors conjectured in Brešar et al. (2009) [1] that the 3-Steiner interval on a connected graph G satisfies the betweenness axiom ( b 2 ) 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 2 k + 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.