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Keywords:
connected graph; signpost system
Summary:
By a signpost system we mean an ordered pair $(W, P)$, where $W$ is a finite nonempty set, $P \subseteq W \times W \times W$ and the following statements hold: $\text{if } (u, v, w) \in P, \text{ then } (v, u, u) \in P\text{ and } (v, u, w) \notin P,\text{ for all }u, v, w \in W; \text{ if } u \ne v,i \text{ then there exists } r \in W \text{ such that } (u, r, v) \in P,\text{ for all } u, v \in W.$ We say that a signpost system $(W, P)$ is smooth if the folowing statement holds for all $u, v, x, y, z \in W$: if $(u, v, x), (u, v, z), (x, y, z) \in P$, then $(u, v, y) \in P$. We say thay a signpost system $(W, P)$ is simple if the following statement holds for all $u, v, x, y \in W$: if $(u, v, x), (x, y, v) \in P$, then $(u, v, y), (x, y, u) \in P$. 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 $u, v \in W$: $u$ and $v$ are adjacent in $G$ if and only if $(u,v, v) \in P$. 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.
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