  The scissors congruence conjecture for the unimodular group is an analogue of
Hilbert's third problem, for the equidecomposability of polytopes. Liu and
Osserman studied the Ehrhart quasi-polynomials of polytopes naturally
associated to graphs whose vertices have degree one or three. In this paper, we
prove the scissors congruence conjecture, posed by Haase and McAllister, for
this class of polytopes. The key ingredient in the proofs is the nearest
neighbor interchange on graphs and a naturally arising piecewise unimodular
transformation.
