## Abstract

Consider the following natural variation of the degree realization problem. Let G= (V, E) be a simple undirected graph of order n. Let f∈R≥0n be a vector of vertex requirements, and let w∈R≥0n be a vector of provided services at the vertices. Then w satisfies f on G if the constraints ∑ _{j}_{∈}_{N}_{(}_{i}_{)}w_{j}= f_{i} are satisfied for all i∈ V, where N(i) denotes the neighborhood of i. Given a requirements vector f, the Weighted Graph Realization problem asks for a suitable graph G and a vector w of provided services that satisfy f on G. In [7] it is observed that any requirement vector where n is even can be realized. If n is odd, the problem becomes much harder. For the unsolved cases, the decision of whether f is realizable or not can be formulated as whether f_{n} (the largest requirement) lies within certain intervals. In [5] some intervals are identified where f can be realized, and their complements form n-32 connected intervals (“unknown domains”) which we give odd indices k= 1, 3, …, n- 4. The unknown domain for k= 1 is shown to be unrealizable. Our main result presents structural properties that a graph must have if it realizes a vector in one of these unknown domains for k≥ 3. The unknown domains are characterized by inequalities which we translate to graph properties. Our analysis identifies several realizable sub-intervals, and shows that each of the unknown domains has at least one sub-interval that cannot be realized.

Original language | English |
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Title of host publication | WALCOM |

Subtitle of host publication | Algorithms and Computation - 16th International Conference and Workshops, WALCOM 2022, Proceedings |

Editors | Petra Mutzel, Md. Saidur Rahman, Slamin |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 315-327 |

Number of pages | 13 |

ISBN (Print) | 9783030967307 |

DOIs | |

State | Published - 2022 |

Event | 16th International Conference and Workshops on Algorithms and Computation, WALCOM 2022 - Jember, Indonesia Duration: 24 Mar 2022 → 26 Mar 2022 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 13174 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 16th International Conference and Workshops on Algorithms and Computation, WALCOM 2022 |
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Country/Territory | Indonesia |

City | Jember |

Period | 24/03/22 → 26/03/22 |

### Bibliographical note

Publisher Copyright:© 2022, Springer Nature Switzerland AG.

### Funding

Supported in part by a US-Israel BSF grant (2018043). Partly supported by ARL 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 |

US-Israel BSF | 2018043 |

Army Research Laboratory |

## Keywords

- Degree sequence
- Graph realization
- Graph-algorithms