## Abstract

The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles. In this paper, we initiate the study of neighborhood degree profiles, wherein, our focus is on the natural problem of realizing maximum neighborhood degrees. More specifically, we ask the following question: “Given a sequence D of n non-negative integers 0 ≤ d1 ≤ · · · ≤ dn, does there exist a simple graph with vertices v1, . . ., vn such that for every 1 ≤ i ≤ n, the maximum degree in the neighborhood of vi is exactly di?” We provide in this work various results for maximum-neighborhood-degree for general n vertex graphs. Our results are first of its kind that studies extremal neighborhood degree profiles. For closed as well as open neighborhood degree profiles, we provide a complete realizability criteria. We also provide tight bounds for the number of maximum neighbouring degree profiles of length n that are realizable. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time.

Original language | English |
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Title of host publication | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 |

Editors | Susanne Albers |

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

ISBN (Electronic) | 9783959771504 |

DOIs | |

State | Published - 1 Jun 2020 |

Event | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 - Torshavn, Faroe Islands Duration: 22 Jun 2020 → 24 Jun 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 162 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 |
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Country/Territory | Faroe Islands |

City | Torshavn |

Period | 22/06/20 → 24/06/20 |

### Bibliographical note

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

## Keywords

- Extremum-degree
- Graph realization
- Neighborhood profile