### Abstract

Special cases of the edge disjoint realizations of two tree degree sequences are considered in this paper. We show that if there is no node which have degree one in both degree sequences, then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. We also show that the edge-disjoint realization problem is in P for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees. On the other hand, we show that the following problem is already NP-complete: given two graphical degree sequences D_{1} and D_{2} such that D_{2} is a tree degree sequence, decide if there exist edge-disjoint realizations of D_{1} and D_{2} where the realization of D_{2} does not need to be a tree. Finally, we show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.

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

Pages (from-to) | 11-18 |

Number of pages | 8 |

Journal | Informatica (Slovenia) |

Volume | 43 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Degree sequences
- Edge-disjoint realizations
- Packing spanning trees
- Uniform sampling

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Computer Science Applications
- Artificial Intelligence

### Cite this

*Informatica (Slovenia)*,

*43*(1), 11-18. https://doi.org/10.31449/inf.v43i1.2675

**Packing tree degree sequences.** / Bérczi, Kristóf; Király, Zoltán; Liu, Changshuo; Miklós, I.

Research output: Contribution to journal › Article

*Informatica (Slovenia)*, vol. 43, no. 1, pp. 11-18. https://doi.org/10.31449/inf.v43i1.2675

}

TY - JOUR

T1 - Packing tree degree sequences

AU - Bérczi, Kristóf

AU - Király, Zoltán

AU - Liu, Changshuo

AU - Miklós, I.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Special cases of the edge disjoint realizations of two tree degree sequences are considered in this paper. We show that if there is no node which have degree one in both degree sequences, then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. We also show that the edge-disjoint realization problem is in P for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees. On the other hand, we show that the following problem is already NP-complete: given two graphical degree sequences D1 and D2 such that D2 is a tree degree sequence, decide if there exist edge-disjoint realizations of D1 and D2 where the realization of D2 does not need to be a tree. Finally, we show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.

AB - Special cases of the edge disjoint realizations of two tree degree sequences are considered in this paper. We show that if there is no node which have degree one in both degree sequences, then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. We also show that the edge-disjoint realization problem is in P for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees. On the other hand, we show that the following problem is already NP-complete: given two graphical degree sequences D1 and D2 such that D2 is a tree degree sequence, decide if there exist edge-disjoint realizations of D1 and D2 where the realization of D2 does not need to be a tree. Finally, we show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.

KW - Degree sequences

KW - Edge-disjoint realizations

KW - Packing spanning trees

KW - Uniform sampling

UR - http://www.scopus.com/inward/record.url?scp=85066745944&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066745944&partnerID=8YFLogxK

U2 - 10.31449/inf.v43i1.2675

DO - 10.31449/inf.v43i1.2675

M3 - Article

AN - SCOPUS:85066745944

VL - 43

SP - 11

EP - 18

JO - Informatica (Slovenia)

JF - Informatica (Slovenia)

SN - 0350-5596

IS - 1

ER -