Title: | On $k$-strong distance in strong digraphs (English) |

Author: | Zhang, Ping |

Language: | English |

Journal: | Mathematica Bohemica |

ISSN: | 0862-7959 (print) |

ISSN: | 2464-7136 (online) |

Volume: | 127 |

Issue: | 4 |

Year: | 2002 |

Pages: | 557-570 |

Summary lang: | English |

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Category: | math |

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Summary: | For a nonempty set $S$ of vertices in a strong digraph $D$, the strong distance $d(S)$ is the minimum size of a strong subdigraph of $D$ containing the vertices of $S$. If $S$ contains $k$ vertices, then $d(S)$ is referred to as the $k$-strong distance of $S$. For an integer $k \ge 2$ and a vertex $v$ of a strong digraph $D$, the $k$-strong eccentricity $\mathop {\mathrm se}_k(v)$ of $v$ is the maximum $k$-strong distance $d(S)$ among all sets $S$ of $k$ vertices in $D$ containing $v$. The minimum $k$-strong eccentricity among the vertices of $D$ is its $k$-strong radius $\mathop {\mathrm srad}_k D$ and the maximum $k$-strong eccentricity is its $k$-strong diameter $_k D$. The $k$-strong center ($k$-strong periphery) of $D$ is the subdigraph of $D$ induced by those vertices of $k$-strong eccentricity $\mathop {\mathrm srad}_k(D)$ ($_k (D)$). It is shown that, for each integer $k \ge 2$, every oriented graph is the $k$-strong center of some strong oriented graph. A strong oriented graph $D$ is called strongly $k$-self-centered if $D$ is its own $k$-strong center. For every integer $r \ge 6$, there exist infinitely many strongly 3-self-centered oriented graphs of 3-strong radius $r$. The problem of determining those oriented graphs that are $k$-strong peripheries of strong oriented graphs is studied. (English) |

Keyword: | strong distance |

Keyword: | strong eccentricity |

Keyword: | strong center |

Keyword: | strong periphery |

MSC: | 05C12 |

MSC: | 05C20 |

idZBL: | Zbl 1003.05037 |

idMR: | MR1942641 |

DOI: | 10.21136/MB.2002.133957 |

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Date available: | 2009-09-24T22:05:07Z |

Last updated: | 2020-07-29 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/133957 |

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Reference: | [1] F. Buckley, Z. Miller, P. J. Slater: On graphs containing a given graph as center.J. Graph Theory 5 (1981), 427–434. MR 0635706, 10.1002/jgt.3190050413 |

Reference: | [2] G. Chartrand, D. Erwin, M. Raines, P. Zhang: Strong distance in strong digraphs.J. Combin. Math. Combin. Comput. 31 (1999), 33–44. MR 1726945 |

Reference: | [3] G. Chartrand, D. Erwin, M. Raines, P. Zhang: On strong distance in strong oriented graphs.Congr. Numer. 141 (1999), 49–63. MR 1744211 |

Reference: | [4] G. Chartrand, L. Lesniak: Graphs $\&$ Digraphs, third edition.Chapman $\&$ Hall, New York, 1996. MR 1408678 |

Reference: | [5] G. Chartrand, O. R. Oellermann, S. Tian, H. B. Zou: Steiner distance in graphs.Čas. Pěst. Mat. 114 (1989), 399–410. MR 1027236 |

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