### Abstract

An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Threshold graphs are special kinds of cographs, which themselves are special kinds of graphs of clique-width 2. We obtain a running time of O(nlog^{2}n) for computing the characteristic polynomial, while the previously fastest algorithm ran in quadratic time. We improve the running time drastically in the case where there is a small number of alternations between 0's and 1's in the sequence defining a threshold graph.

Original language | English (US) |
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Pages (from-to) | 3-10 |

Number of pages | 8 |

Journal | Theoretical Computer Science |

Volume | 657 |

DOIs | |

State | Published - Jan 2 2017 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

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**Efficient computation of the characteristic polynomial of a threshold graph.** / Fürer, Martin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Efficient computation of the characteristic polynomial of a threshold graph

AU - Fürer, Martin

PY - 2017/1/2

Y1 - 2017/1/2

N2 - An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Threshold graphs are special kinds of cographs, which themselves are special kinds of graphs of clique-width 2. We obtain a running time of O(nlog2n) for computing the characteristic polynomial, while the previously fastest algorithm ran in quadratic time. We improve the running time drastically in the case where there is a small number of alternations between 0's and 1's in the sequence defining a threshold graph.

AB - An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Threshold graphs are special kinds of cographs, which themselves are special kinds of graphs of clique-width 2. We obtain a running time of O(nlog2n) for computing the characteristic polynomial, while the previously fastest algorithm ran in quadratic time. We improve the running time drastically in the case where there is a small number of alternations between 0's and 1's in the sequence defining a threshold graph.

UR - http://www.scopus.com/inward/record.url?scp=84979650049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84979650049&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2016.07.013

DO - 10.1016/j.tcs.2016.07.013

M3 - Article

AN - SCOPUS:84979650049

VL - 657

SP - 3

EP - 10

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -