## Abstract

The closed neighbourhood N_{G}[X] of a vertex x in a graph G is the subgraph of G induced by x and all neighbours of x. The seed of a vertex x ∈ G is the subgraph of G induced by all vertices of G\N_{G}[x] and we denote it by S_{G}(x). A graph F is a seed graph if there exists a graph G such that S_{G}(x) ≅ F for each x ∈ G. In this paper seed graphs with more than two components are studied. It is shown that if all components are of equal order, size or regularity then they are all isomorphic to a complete graph. In the general case it is shown how the structure of any component F_{i} of a seed graph F depends on the structure of all components 'smaller' than F_{i} in the sense of 'smaller order', 'smaller size' or 'smaller degree' in the case of regular components.

Original language | English (US) |
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Pages (from-to) | 115-126 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 233 |

Issue number | 1-3 |

DOIs | |

State | Published - Apr 28 2001 |

### Bibliographical note

Funding Information:This work was in part supported by GACR" Grant No. 201=98=1451. ∗Tel.: +420-69-699-3496; fax: +420-69-691-9597. E-mail address: dalibor.froncek@vsb.cz (D. Fronc"ek).

## Keywords

- Isomorphic survivor graphs
- Local properties of graphs
- Seed graphs