  It is known that, if a locally perturbed periodic self-adjoint operator on a
combinatorial or quantum graph admits an eigenvalue embedded in the continuous
spectrum, then the associated eigenfunction is compactly supported--that is, if
the Fermi surface is irreducible, which occurs generically in dimension two or
higher. This article constructs a class of operators whose Fermi surface is
reducible for all energies by coupling several periodic systems. The components
of the Fermi surface correspond to decoupled spaces of hybrid states, and in
certain frequency bands, some components contribute oscillatory hybrid states
(corresponding to spectrum) and other components contribute only exponential
ones. This separation allows a localized defect to suppress the oscillatory
(radiation) modes and retain the evanescent ones, thereby leading to embedded
eigenvalues whose associated eigenfunctions decay exponentially but are not
compactly supported.
