Strict Betweennesses Induced by Posets as well as by Graphs

For a finite poset P = (V, ≤ ), let ${\cal B}_s(P)$ consist of all triples (x,y,z) ∈ V3 such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let ${\cal B}_s(G)$ consist of all triples (x,y,z) ∈ V3 such that y is an internal vertex on an induced path in G between x and z. The ternary relations ${\cal B}_s(P)$ and ${\cal B}_s(G)$ 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 ${\cal B}_s(P)={\cal B}_s(G)$.