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**Intrinsic linking in directed graphs.**
*(English)*
Zbl 1337.57016

This work extends the notion of intrinsic linking to directed graphs. A directed graph \(G\) is called intrinsically linked if it contains a nontrivial link consisting of a pair of consistently oriented directed cycles in every spatial embedding. The double directed version of a graph \(G\), denoted \(\overline{D(G)}\), is obtained by replacing each edge \(e\) of \(G\) by a pair of directed edges with opposite orientations joining the same pair of vertices as \(e\). The main result of this paper is that the double directed version of a graph \(G\) is intrinsically linked if and only if \(G\) is intrinsically linked.

As a corollary to this result it is shown that the complete symmetric directed graph on six vertices, \(\overline {J_6}\), is intrinsically linked, thus extending the work of Conway, Gordon and Sachs [J. H. Conway and C. McA. Gordon, J. Graph Theory 7, 445–453 (1983; Zbl 0524.05028)], [H. Sachs, in: Finite and infinite sets, 6th Hung. Combin. Colloq., Eger/Hung. 1981, Vol. II, Colloq. Math. Soc. János Bolyai 37, 649–662 (1984; Zbl 0568.05026)]. Several other results are shown for \(\overline {J_6}\) including the number of allowable edges that can be deleted to make the subgraph intrinsically linked or not.

As a corollary to this result it is shown that the complete symmetric directed graph on six vertices, \(\overline {J_6}\), is intrinsically linked, thus extending the work of Conway, Gordon and Sachs [J. H. Conway and C. McA. Gordon, J. Graph Theory 7, 445–453 (1983; Zbl 0524.05028)], [H. Sachs, in: Finite and infinite sets, 6th Hung. Combin. Colloq., Eger/Hung. 1981, Vol. II, Colloq. Math. Soc. János Bolyai 37, 649–662 (1984; Zbl 0568.05026)]. Several other results are shown for \(\overline {J_6}\) including the number of allowable edges that can be deleted to make the subgraph intrinsically linked or not.

Reviewer: Lew Ludwig (Granville)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M15 | Relations of low-dimensional topology with graph theory |

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

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\textit{J. S. Foisy} et al., Osaka J. Math. 52, No. 3, 817--831 (2015; Zbl 1337.57016)

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### References:

[1] | J. Bang-Jensen and G. Gutin: Digraphs, second edition, Springer Monographs in Mathematics, Springer, London, 2009. |

[2] | C. Binucci, W. Didimo and F. Giordano: Maximum upward planar subgraphs of embedded planar digraphs , Comput. Geom. 41 (2008), 230-246. · Zbl 1152.05319 · doi:10.1016/j.comgeo.2008.02.001 |

[3] | G. Bowlin and J. Foisy: Some new intrinsically 3-linked graphs , J. Knot Theory Ramifications 13 (2004), 1021-1027. · Zbl 1071.57003 · doi:10.1142/S0218216504003652 |

[4] | J.H. Conway and C.McA. Gordon: Knots and links in spatial graphs , J. Graph Theory 7 (1983), 445-453. · Zbl 0524.05028 · doi:10.1002/jgt.3190070410 |

[5] | E. Flapan, R. Naimi and J. Pommersheim: Intrinsically triple linked complete graphs , Topology Appl. 115 (2001), 239-246. · Zbl 0988.57003 · doi:10.1016/S0166-8641(00)00064-X |

[6] | P.P. Howards, E.F. Schisterman, C. Poole, J.S. Kaufman and C.R. Weinberg: “Toward a clearer definition of confounding” revisited with directed acyclic graphs , American Journal of Epidemiology 176 (2012), 506-511. |

[7] | R. Motwani, A. Raghunathan and H. Saran: Constructive results from graph minors: lintless embeddings ; in 29th Annual Symposium on Foundations of Computer Science (White Plains, NY, 1988), IEEE, 1988, 398-409. |

[8] | N. Robertson and P.D. Seymour: Graph minors . XX. Wagner’s conjecture , J. Combin. Theory Ser. B 92 (2004), 325-357. · Zbl 1061.05088 · doi:10.1016/j.jctb.2004.08.001 |

[9] | N. Robertson, P.D. Seymour and R. Thomas: Sachs’ linkless embedding conjecture , J. Combin. Theory Ser. B 64 (1995), 185-227. · Zbl 0832.05032 · doi:10.1006/jctb.1995.1032 |

[10] | H. Sachs: On spatial representations of finite graphs ; in Finite and Infinite Sets, I, II (Eger, 1981), Colloq. Math. Soc. János Bolyai 37 , North-Holland, Amsterdam, 1984, 649-662. |

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