### Abstract

Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d (v) ≥ a (v) + b (v) + 1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A, B) such that d_{A} (v) ≥ a (v) for every v ∈ A and d_{B} (v) ≥ b (v) for every v ∈ B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d (v) ≥ a + b (a, b ≥ 1) or just d (v) ≥ a + b - 1 (a, b ≥ 2) if G contains no cycles shorter than 4 or 5, respectively. The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.

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

Pages (from-to) | 979-988 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 155 |

Issue number | 8 |

DOIs | |

Publication status | Published - Apr 15 2007 |

### Fingerprint

### Keywords

- Graph decomposition
- Polynomial algorithm
- Vertex degree
- Vertex partition

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Applied Mathematics*,

*155*(8), 979-988. https://doi.org/10.1016/j.dam.2006.10.005

**Efficient algorithms for decomposing graphs under degree constraints.** / Bazgan, Cristina; Tuza, Z.; Vanderpooten, Daniel.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 155, no. 8, pp. 979-988. https://doi.org/10.1016/j.dam.2006.10.005

}

TY - JOUR

T1 - Efficient algorithms for decomposing graphs under degree constraints

AU - Bazgan, Cristina

AU - Tuza, Z.

AU - Vanderpooten, Daniel

PY - 2007/4/15

Y1 - 2007/4/15

N2 - Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d (v) ≥ a (v) + b (v) + 1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A, B) such that dA (v) ≥ a (v) for every v ∈ A and dB (v) ≥ b (v) for every v ∈ B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d (v) ≥ a + b (a, b ≥ 1) or just d (v) ≥ a + b - 1 (a, b ≥ 2) if G contains no cycles shorter than 4 or 5, respectively. The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.

AB - Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d (v) ≥ a (v) + b (v) + 1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A, B) such that dA (v) ≥ a (v) for every v ∈ A and dB (v) ≥ b (v) for every v ∈ B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d (v) ≥ a + b (a, b ≥ 1) or just d (v) ≥ a + b - 1 (a, b ≥ 2) if G contains no cycles shorter than 4 or 5, respectively. The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.

KW - Graph decomposition

KW - Polynomial algorithm

KW - Vertex degree

KW - Vertex partition

UR - http://www.scopus.com/inward/record.url?scp=33947270327&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947270327&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2006.10.005

DO - 10.1016/j.dam.2006.10.005

M3 - Article

AN - SCOPUS:33947270327

VL - 155

SP - 979

EP - 988

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 8

ER -