An algebraic characterization of geodetic graphs

We say that a binary operation * is associated with a (finite undirected) graph G (without loops and multiple edges) if * is defined on V(G) and uv ∈ E(G) if and only if u ≠ v, u * v = v and v * u = u for any u, v ∈ V(G). In the paper it is proved that a connected graph G is geodetic if and only if there exists a binary operation associated with G which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).