Cutsets and EF1 Fair Division of Graphs

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.