Travel groupoids
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 659-675.

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In this paper, by a travel groupoid is meant an ordered pair $(V, *)$ such that $V$ is a nonempty set and $*$ is a binary operation on $V$ satisfying the following two conditions for all $u, v \in V$: \[ (u * v) * u = u; \text{ if }(u * v ) * v = u, \text{ then } u = v. \] Let $(V, *)$ be a travel groupoid. It is easy to show that if $x, y \in V$, then $x * y = y$ if and only if $y * x = x$. We say that $(V, *)$ is on a (finite or infinite) graph $G$ if $V(G) = V$ and \[ E(G) = \lbrace \lbrace u, v\rbrace \: u, v \in V \text{ and } u \ne u * v = v\rbrace . \] Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.
Classification : 05C12, 05C25, 05C38, 20N02
Mots-clés : travel groupoid; graph; path; geodetic graph 
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Nebeský, Ladislav. Travel groupoids. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 659-675. https://geodesic-test.mathdoc.fr/item/CMJ_2006__56_2_a29/