Strict Betweennesses Induced by Posets as well as by Graphs

For a finite poset P = (V,≤), let Bs(P ) consist of all triples (x, y, z) ∈ V 3 such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let Bs(G) consist of all triples (x, y, z) ∈ V 3 such that y is an internal vertex on an induced path in G between x and z. The ternary relations Bs(P ) and Bs(G) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation Bs(P ) = Bs(G).