## Abstract

Gerke et al. (2019) introduced a game-theoretic model to study public good provision in social networks when there are constraints on sharing. This model generates a purely graph-theoretic problem termed exact capacitated domination. In the problem we are given a capacitated graph, a graph with a parameter defined on each vertex that is interpreted as the capacity of that vertex. The objective is to find a DP-Nash subgraph: a spanning bipartite subgraph with partite sets D and P, called the D-set and P-set respectively, such that no vertex in P is isolated and that each vertex in D is adjacent to a number of vertices equal to its capacity. We show that whether a capacitated graph has a unique DP-Nash subgraph can be decided in polynomial time. However, we also show that the closely related problem of deciding whether a capacitated graph has a unique D-set is co-NP-complete.

Originalsprog | Engelsk |
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Tidsskrift | Discrete Applied Mathematics |

Vol/bind | 332 |

Sider (fra-til) | 155-169 |

ISSN | 0166-218X |

DOI | |

Status | Udgivet - 15. jun. 2023 |

### Bibliografisk note

Funding Information:Anders Yeo’s research was partially supported by grant DFF-7014-00037B of Independent Research Fund Denmark .

Publisher Copyright:

© 2023 The Author(s)