On signpost systems and connected graphs

Abstract

By a signpost system we mean an ordered pair (W, P), where W is a finite nonempty set, P $\subseteq$ W × W × W and the following statements hold: if (u, v, w) ∈ P, then (v, u, u) ∈ P and (v, u, w) ∉ P, for all u, v, w ∈ W; if u ≠ v; then there exists r ∈ W such that (u, r, v) ∈ P, for all u, v ∈ W. We say that a signpost system (W, P) is smooth if the folowing statement holds for all u, v, x, y, z ∈ W: if (u, v, x), (u, v, z), (x, y, z) ∈ P, then (u, v, y) ∈ P. We say thay a signpost system (W, P) is simple if the following statement holds for all u, v, x, y ∈ W: if (u, v, x), (x, y, v) ∈ P, then (u, v, y), (x, y, u) ∈ P.

By the underlying graph of a signpost system (W, P) we mean the graph G with V(G) = W and such that the following statement holds for all distinct u, v ∈ W: u and v are adjacent in G if and only if (u, v, v) ∈ P. The main result of this paper is as follows: If G is a graph, then the following three statements are equivalent: G is connected; G is the underlying graph of a simple smooth signpost system; G is the underlying graph of a smooth signpost system.