We consider the problem of fairly allocating the vertices of a graph among $n$ agents, where the value of a bundle is determined by its cut value -- the number of edges with exactly one endpoint in the bundle. This model naturally captures applications such as team formation and network partitioning, where valuations are inherently non-monotonic: the marginal values may be positive, negative, or zero depending on the composition of the bundle. We focus on the fairness notion of envy-freeness up to one item (EF1) and explore its compatibility with several efficiency concepts such as Transfer Stability (TS) that prohibits single-item transfers that benefit one agent without making the other worse-off. For general graphs, our results uncover a non-monotonic relationship between the number of agents $n$ and the existence of allocations satisfying EF1 and transfer stability (TS): such allocations always exist for $n=2$, may fail to exist for $n=3$, but exist again for all $n\geq 4$. We further show that existence can be guaranteed for any $n$ by slightly weakening the efficiency requirement or by restricting the graph to forests. All of our positive results are achieved via efficient algorithms.

翻译：我们考虑将图的顶点公平分配给 $n$ 个智能体的问题，其中捆绑包的价值由其割值决定——即恰好有一个端点位于该捆绑包内的边的数量。该模型自然地捕捉了团队组建和网络划分等应用场景，其中估值本质上具有非单调性：边际价值可能为正、负或零，具体取决于捆绑包的构成。我们聚焦于无嫉妒性至多一项（EF1）这一公平性概念，并探讨其与多种效率概念的兼容性，例如禁止单物品转移使一方受益而不使另一方受损的转移稳定性（TS）。对于一般图，我们的结果揭示了智能体数量 $n$ 与满足 EF1 和转移稳定性（TS）的分配存在性之间的非单调关系：当 $n=2$ 时此类分配始终存在，当 $n=3$ 时可能不存在，但当 $n\\geq 4$ 时又始终存在。我们进一步证明，通过略微弱化效率要求或将图限制为森林，可以保证对任意 $n$ 均存在满足条件的分配。我们所有的肯定性结果均通过高效算法实现。