The 2-linkage problem for acyclic digraphs.

*(English)*Zbl 0563.05027The paper solves the problem of finding two disjoint directed paths with prescribed ends in an acyclic digraph. This is accomplished by reducing the problem to one of planarity test of an undirected graph. Necessary and sufficient conditions are given in terms of certain forbidden subdigraphs. As a result, it is shown that if any three cycles of a digraph have a common vertex, then all cycles have a common vertex. A precise structural characterization of digraphs of order n and girth \(g\geq 2n/3\) is given. From this it is shown that any such digraph has at most \(3^{n-g}\) distinct cycles and that a digraph of order n and girth \(g\geq 4\) has at most \(2^{n-g}\) shortest cycles.

Reviewer: W.-K.Chen

##### MSC:

05C20 | Directed graphs (digraphs), tournaments |

05C38 | Paths and cycles |

05C10 | Planar graphs; geometric and topological aspects of graph theory |

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##### References:

[1] | Allender, E., On the number of cycles possible in digraphs with large girths, Discrete appl. math., 10, 211-225, (1985), preprint · Zbl 0565.05049 |

[2] | Bermond, J.-C.; Thomassen, C., Cycles in digraphs—a survey, J. graph theory, 5, 1-43, (1981) · Zbl 0458.05035 |

[3] | Birkhoff, G., Lattice theory, (1940), Amer. Math. Soc. New York · Zbl 0126.03801 |

[4] | Caccetta, L.; Häggkvist, R., On minimal digraphs with given girth, (), 181-187 · Zbl 0406.05033 |

[5] | Fortune, S.; Hopcroft, J.; Wyllie, J., The directed subgraph homeomorphism problem, Theoret. comput. sci., 10, 111-121, (1980) · Zbl 0419.05028 |

[6] | Hopcroft, J.E.; Tarjan, R.E., Efficient planarity testing, J. ACM, 21, 549-568, (1974) · Zbl 0307.68025 |

[7] | Kosaraju, S.R., On independent circuits of a digraph, J. graph theory, 1, 379-382, (1977) · Zbl 0383.05020 |

[8] | Seymour, P.D., Disjoint paths in graphs, Discrete math., 29, 293-309, (1980) · Zbl 0457.05043 |

[9] | Thomassen, C., 2-linked graphs, Europ. J. combin., 1, 371-378, (1980) · Zbl 0457.05044 |

[10] | Thomassen, C., Planarity and duality of finite and infinite graphs, J. combin. theory ser. B, 29, 244-271, (1980) · Zbl 0441.05023 |

[11] | Thomassen, C., Disjoint cycles in digraphs, Combinatorica, 3, 393-396, (1983) · Zbl 0527.05036 |

[12] | C. Thomassen, Even cycles in digraphs, Europ. J. Combin., to appear. · Zbl 0395.05040 |

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