## Abstract

We consider the problem of realizable interval-sequences. An interval sequence comprises of n integer intervals [ai, bi] such that 0 ≤ ai ≤ bi ≤ n − 1, and is said to be graphic/realizable if there exists a graph with degree sequence, say, D = (d1, . . ., dn) satisfying the condition ai ≤ di ≤ bi, for each i ∈ [1, n]. There is a characterisation (also implying an O(n) verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdös-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in o(n^{2}) time. In this paper, we provide an O(n log n) time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length p non-graphic sequence to a graphic one.

Original language | English |
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Title of host publication | 30th International Symposium on Algorithms and Computation, ISAAC 2019 |

Editors | Pinyan Lu, Guochuan Zhang |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771306 |

DOIs | |

State | Published - Dec 2019 |

Event | 30th International Symposium on Algorithms and Computation, ISAAC 2019 - Shanghai, China Duration: 8 Dec 2019 → 11 Dec 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 149 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 30th International Symposium on Algorithms and Computation, ISAAC 2019 |
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Country/Territory | China |

City | Shanghai |

Period | 8/12/19 → 11/12/19 |

### Bibliographical note

Publisher Copyright:© Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz; licensed under Creative Commons License CC-BY

### Funding

Funding US-Israel BSF grant 2018043; Army Research Laboratory Cooperative Grant, ARL Network Science CTA, W911NF-09-2-0053. US-Israel BSF grant 2018043; Army Research Laboratory Cooperative Grant, ARL Network Science CTA, W911NF-09- 2-0053.

Funders | Funder number |
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ARL Network Science CTA | W911NF-09-2-0053 |

Bloom's Syndrome Foundation | 2018043 |

Army Research Laboratory | |

United States-Israel Binational Science Foundation |

## Keywords

- Graph realization
- Graphic sequence
- Interval sequence