Distance-hereditary graphs and signpost systems

An ordered pair (U,R) is called a signpost system if U is a finite nonempty set, R⊆U×U×U, and the following axioms hold for all u,v,w∈U: (1) if (u,v,w)∈R, then (v,u,u)∈R; (2) if (u,v,w)∈R, then (v,u,w)∉R; (3) if u≠v, then there exists t∈U such that (u,t,v)∈R. (If F is a (finite) connected graph with vertex set U and distance function d, then U together with the set of all ordered triples (u,v,w) of vertices in F such that d(u,v)=1 and d(v,w)=d(u,w)−1 is an example of a signpost system). If (U,R) is a signpost system and G is a graph, then G is called the underlying graph of (U,R) if V(G)=U and xy∈E(G) if and only if (x,y,y)∈R (for all x,y∈U). It is possible to say that a signpost system shows a way how to travel in its underlying graph. The following result is proved: Let (U,R) be a signpost system and let G denote the underlying graph of (U,R). Then G is connected and every induced path in G is a geodesic in G if and only if (U,R) satisfies axioms (4)-(8) stated in this paper; note that axioms (4)-(8)-similarly as axioms (1)-(3)-can be formulated in the language of the first-order logic.