### Abstract

A graph G is called preperfect if each induced subgraph G′ ⊆ G of order at least 2 has two vertices x, y such that either all maximum cliques of G′ containing x contain y, or all maximum independent sets of G′ containing y contain x, too. Giving a partial answer to a problem of Hammer and Maffray [Combinatorica 13 (1993), 199-208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations: (i) A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. (ii) If a graph G is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if G is bipartite, 3-edge-connected, regular of degree d for some d ≥ 3, and contains no 3-edge-connected d′-regular subgraph for any 3 ≤ d′ <d.

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

Pages (from-to) | 759-773 |

Number of pages | 15 |

Journal | Graphs and Combinatorics |

Volume | 17 |

Issue number | 4 |

Publication status | Published - 2001 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*17*(4), 759-773.

**Minimally non-preperfect graphs of small maximum degree.** / Tuza, Z.; Wagler, Annegret.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 17, no. 4, pp. 759-773.

}

TY - JOUR

T1 - Minimally non-preperfect graphs of small maximum degree

AU - Tuza, Z.

AU - Wagler, Annegret

PY - 2001

Y1 - 2001

N2 - A graph G is called preperfect if each induced subgraph G′ ⊆ G of order at least 2 has two vertices x, y such that either all maximum cliques of G′ containing x contain y, or all maximum independent sets of G′ containing y contain x, too. Giving a partial answer to a problem of Hammer and Maffray [Combinatorica 13 (1993), 199-208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations: (i) A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. (ii) If a graph G is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if G is bipartite, 3-edge-connected, regular of degree d for some d ≥ 3, and contains no 3-edge-connected d′-regular subgraph for any 3 ≤ d′

AB - A graph G is called preperfect if each induced subgraph G′ ⊆ G of order at least 2 has two vertices x, y such that either all maximum cliques of G′ containing x contain y, or all maximum independent sets of G′ containing y contain x, too. Giving a partial answer to a problem of Hammer and Maffray [Combinatorica 13 (1993), 199-208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations: (i) A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. (ii) If a graph G is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if G is bipartite, 3-edge-connected, regular of degree d for some d ≥ 3, and contains no 3-edge-connected d′-regular subgraph for any 3 ≤ d′

UR - http://www.scopus.com/inward/record.url?scp=19544391068&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=19544391068&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:19544391068

VL - 17

SP - 759

EP - 773

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 4

ER -