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Route systems of a connected graph. (English) Zbl 0820.05021
For a graph $$G$$, a route system is a set of paths satisfying five axioms, which generalise some properties of the set $${\mathcal S} (G)$$ of all shortest paths of $$G$$. Continuing his previous work (see [Czech. Math. J. 41(116), No. 2, 260-264 (1991; Zbl 0765.05060)] and [Math. Bohem. 119, No. 1, 15-20 (1994; Zbl 0807.05045)]), the author establishes some properties concerning route systems, in particular those minimal or maximal with respect to set theoretical inclusion among all route systems. He shows that (i) a route system $${\mathcal K}$$ is minimal if and only if it satisfies a further axiom; (ii) if $$G$$ is geodetic then $${\mathcal S} (G)$$ is not maximal, unless $$G$$ is a tree; (iii) if $$G$$ is bipartite then $${\mathcal S} (G)$$ is maximal. It is conjectured that the converse of (iii) is true.

##### MSC:
 05C12 Distance in graphs 05C38 Paths and cycles
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