In fair division of a connected graph $G = (V, E)$, each of $n$ agents receives a share of $G$'s vertex set $V$. These shares partition $V$, with each share required to induce a connected subgraph. Agents use their own valuation functions to determine the non-negative numerical values of the shares, which determine whether the allocation is fair in some specified sense. We introduce forbidden substructures called graph cutsets, which block divisions that are fair in the EF1 (envy-free up to one item) sense by cutting the graph into "too many pieces". Two parameters - gap and valence - determine blocked values of $n$. If $G$ guarantees connected EF1 allocations for $n$ agents with valuations that are CA (common and additive), then $G$ contains no elementary cutset of gap $k \ge 2$ and valence in the interval $\[n - k + 1, n - 1\]$. If $G$ guarantees connected EF1 allocations for $n$ agents with valuations in the broader CM (common and monotone) class, then $G$ contains no cutset of gap $k \ge 2$ and valence in the interval $\[n - k + 1, n - 1\]$. These results rule out the existence of connected EF1 allocations in a variety of situations. For some graphs $G$ we can, with help from some new positive results, pin down $G$'s spectrum - the list of exactly which values of $n$ do/do not guarantee connected EF1 allocations. Examples suggest a conjectured common spectral pattern for all graphs. Further, we show that it is NP-hard to determine whether a graph admits a cutset. We also provide an example of a (non-traceable) graph on eight vertices that has no cutsets of gap $\ge 2$ at all, yet fails to guarantee connected EF1 allocations for three agents with CA preferences.

翻译：在连通图 $G = (V, E)$ 的公平分配问题中，$n$ 个智能体各获得图 $G$ 的顶点集 $V$ 的一个子集份额。这些份额构成 $V$ 的一个划分，且每个份额需诱导出连通子图。智能体使用各自的估值函数确定份额的非负数值，从而判定分配在特定意义下是否公平。我们引入一种称为图分割（cutset）的禁止子结构，这类结构通过将图分割成"过多片段"来阻碍满足EF1（无嫉妒至多一项）意义上的公平分配。两个参数——间隙（gap）与价（valence）——决定了被阻碍的智能体数量 $n$ 的取值。若图 $G$ 能保证对具有公共可加（CA）估值的 $n$ 个智能体存在连通EF1分配，则 $G$ 不含间隙 $k \ge 2$ 且价位于区间 $[n - k + 1, n - 1]$ 的基本分割。若图 $G$ 能保证对具有更广泛的公共单调（CM）估值的 $n$ 个智能体存在连通EF1分配，则 $G$ 不含间隙 $k \ge 2$ 且价位于区间 $[n - k + 1, n - 1]$ 的分割。这些结果排除了多种情形下连通EF1分配的存在性。借助若干新的正面结论，对于某些图 $G$，我们可以确定其频谱（spectrum）——即精确列出能/不能保证连通EF1分配的 $n$ 值列表。示例表明所有图可能具有相同的谱模式。此外，我们证明判定一个图是否存在分割是NP-难问题。我们还给出了一个八顶点（非可迹）图的例子：该图完全不含间隙 $\ge 2$ 的分割，却无法保证对具有CA偏好的三个智能体存在连通EF1分配。