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 x y ∈ 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.