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travel groupoid; geodetic graph; infinite graph
The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set $V$ and a binary operation $*$ on $V$ satisfying two axioms. We can associate a graph with a travel groupoid. We say that a graph $G$ has a travel groupoid if the graph associated with the travel groupoid is equal to $G$. Nebeský gave a characterization of finite graphs having a travel groupoid. In this paper, we study travel groupoids on infinite graphs. We answer a question posed by Nebeský, and we also give a characterization of infinite graphs having a travel groupoid.
[1] Nebeský, L.: An algebraic characterization of geodetic graphs. Czech. Math. J. 48 (1998), 701-710. DOI 10.1023/A:1022435605919 | MR 1658245 | Zbl 0949.05022
[2] Nebeský, L.: A tree as a finite nonempty set with a binary operation. Math. Bohem. 125 (2000), 455-458. MR 1802293 | Zbl 0963.05032
[3] Nebeský, L.: New proof of a characterization of geodetic graphs. Czech. Math. J. 52 (2002), 33-39. DOI 10.1023/A:1021715219620 | MR 1885455 | Zbl 0995.05124
[4] Nebeský, L.: On signpost systems and connected graphs. Czech. Math. J. 55 (2005), 283-293. DOI 10.1007/s10587-005-0022-0 | MR 2137138 | Zbl 1081.05054
[5] Nebeský, L.: Travel groupoids. Czech. Math. J. 56 (2006), 659-675. DOI 10.1007/s10587-006-0046-0 | MR 2291765 | Zbl 1157.20336
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