## 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.

Original language | English |
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Journal | Discrete Applied Mathematics |

Volume | 332 |

Pages (from-to) | 155-169 |

ISSN | 0166-218X |

DOIs | |

Publication status | Published - 15. Jun 2023 |

## Keywords

- D-set
- Exact capacitated domination
- Nash subgraph