# Article

 Title: A characterization of the set of all shortest paths in a connected graph (English) Author: Nebeský, Ladislav Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 119 Issue: 1 Year: 1994 Pages: 15-20 Summary lang: English . Category: math . Summary: Let $G$ be a (finite undirected) connected graph (with no loop or multiple edge). The set $\Cal L$ of all shortest paths in $G$ is defined as the set of all paths $\xi$, then the lenght of $\xi$ does not exceed the length of $\varsigma$. While the definition of $\Cal L$ is based on determining the length of a path. Theorem 1 gives - metaphorically speaking - an "almost non-metric" characterization of $\Cal L$: a characterization in which the length of a path greater than one is not considered. Two other theorems are derived from Theorem 1. One of them (Theorem 3) gives a characterization of geodetic graphs. (English) Keyword: geodetic graphs Keyword: connected graph Keyword: shortest paths MSC: 05C12 MSC: 05C38 MSC: 05C75 idZBL: Zbl 0807.05045 idMR: MR1303548 DOI: 10.21136/MB.1994.126208 . Date available: 2009-09-24T21:02:18Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/126208 . Reference: [1] M. Behzad G. Chartrand, L. Lesniak-Foster: Graphs & Digraphs.Prindle, Weber & Schmidt, Boston, 1979. MR 0525578 Reference: [2] D.C. Kay, G. Chartrand: A characterization of certain ptolemaic graphs.Canad. J. Math. 17 (1965), 342-346. Zbl 0139.17301, MR 0175113, 10.4153/CJM-1965-034-0 Reference: [3] H.M. Mulder: The Interval Function of a Graph.Mathematisch Centrum, Amsterdam, 1980. Zbl 0446.05039, MR 0605838 Reference: [4] L. Nebeský: Route systems and bipartite graphs.Czechoslovak Math. Journal 41 (116) (1991), 260-264. MR 1105440 .

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