### Abstract

In a connected graph define the k‐center as the set of vertices whose distance from any other vertex is at most k. We say that a vertex set S d‐dominates G if for every vertex x there is a y ∈ S whose distance from x is at most d. Call a graph P_{t}‐free if it does not contain a path on t vertices as an induced subgraph. We prove that a connected graph is P_{2k‐1}‐free (P_{2k}‐free) if and only if each of its connected induced subgraphs H satisfy the following property: The k‐center of H (k ‐ 1)‐dominates ((k ‐ 2)‐dominates) H. Moreover, we show that the subgraph induced by the (t ‐ 3)‐center in any P_{t}‐free connected graph is again connected and has diameter at most t ‐ 3.

Original language | English |
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Pages (from-to) | 455-464 |

Number of pages | 10 |

Journal | Journal of Graph Theory |

Volume | 14 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1990 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*14*(4), 455-464. https://doi.org/10.1002/jgt.3190140409