### Abstract

The reduction number r(G) of a graph G is the maximum integer m≤|E(G)| such that the graphs G-E, ⊆E(G),|E|≤m, are mutually non-isomorphic, i.e., each graph is unique as a subgraph of G. We prove that [InlineMediaObject not available: see fulltext.] and show by probabilistic methods that r(G) can come close to this bound for large orders. By direct construction, we exhibit graphs with large reduction number, although somewhat smaller than the upper bound. We also discuss similarities to a parameter introduced by Erdos and Rényi capturing the degree of asymmetry of a graph, and we consider graphs with few circuits in some detail.

Original language | English |
---|---|

Pages (from-to) | 453-470 |

Number of pages | 18 |

Journal | Graphs and Combinatorics |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2006 |

### Fingerprint

### Keywords

- Asymmetry
- Edge deletion
- Isomorphism
- Random graph
- Reduction number
- Spanning subgraph
- Symmetry
- Tree
- Unicyclic graph
- Unique subgraph

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*22*(4), 453-470. https://doi.org/10.1007/s00373-006-0676-x

**Largest non-unique subgraphs.** / Andersen, Lars Døvling; Vestergaard, Preben Dahl; Tuza, Z.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 22, no. 4, pp. 453-470. https://doi.org/10.1007/s00373-006-0676-x

}

TY - JOUR

T1 - Largest non-unique subgraphs

AU - Andersen, Lars Døvling

AU - Vestergaard, Preben Dahl

AU - Tuza, Z.

PY - 2006/12

Y1 - 2006/12

N2 - The reduction number r(G) of a graph G is the maximum integer m≤|E(G)| such that the graphs G-E, ⊆E(G),|E|≤m, are mutually non-isomorphic, i.e., each graph is unique as a subgraph of G. We prove that [InlineMediaObject not available: see fulltext.] and show by probabilistic methods that r(G) can come close to this bound for large orders. By direct construction, we exhibit graphs with large reduction number, although somewhat smaller than the upper bound. We also discuss similarities to a parameter introduced by Erdos and Rényi capturing the degree of asymmetry of a graph, and we consider graphs with few circuits in some detail.

AB - The reduction number r(G) of a graph G is the maximum integer m≤|E(G)| such that the graphs G-E, ⊆E(G),|E|≤m, are mutually non-isomorphic, i.e., each graph is unique as a subgraph of G. We prove that [InlineMediaObject not available: see fulltext.] and show by probabilistic methods that r(G) can come close to this bound for large orders. By direct construction, we exhibit graphs with large reduction number, although somewhat smaller than the upper bound. We also discuss similarities to a parameter introduced by Erdos and Rényi capturing the degree of asymmetry of a graph, and we consider graphs with few circuits in some detail.

KW - Asymmetry

KW - Edge deletion

KW - Isomorphism

KW - Random graph

KW - Reduction number

KW - Spanning subgraph

KW - Symmetry

KW - Tree

KW - Unicyclic graph

KW - Unique subgraph

UR - http://www.scopus.com/inward/record.url?scp=33846089430&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846089430&partnerID=8YFLogxK

U2 - 10.1007/s00373-006-0676-x

DO - 10.1007/s00373-006-0676-x

M3 - Article

AN - SCOPUS:33846089430

VL - 22

SP - 453

EP - 470

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 4

ER -