  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.
