Minimally non-preperfect graphs of small maximum degree

Z. Tuza, Annegret Wagler

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)759-773
Number of pages15
JournalGraphs and Combinatorics
Volume17
Issue number4
Publication statusPublished - 2001

Fingerprint

Hammers
Maximum Degree
Odd Cycle
Line Graph
Graph in graph theory
If and only if
Maximum Clique
Maximum Independent Set
Induced Subgraph
Bipartite Graph
3D
Connected graph
Subgraph
Complement
Partial
Cycle

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics

Cite this

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

In: Graphs and Combinatorics, Vol. 17, No. 4, 2001, p. 759-773.

Research output: Contribution to journalArticle

Tuza, Z. ; Wagler, Annegret. / Minimally non-preperfect graphs of small maximum degree. In: Graphs and Combinatorics. 2001 ; Vol. 17, No. 4. pp. 759-773.
@article{1e8ec918919d4bd3a76b3819f5d2a9d7,
title = "Minimally non-preperfect graphs of small maximum degree",
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′",
author = "Z. Tuza and Annegret Wagler",
year = "2001",
language = "English",
volume = "17",
pages = "759--773",
journal = "Graphs and Combinatorics",
issn = "0911-0119",
publisher = "Springer Japan",
number = "4",

}

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 -