Cubic graphs, their Ehrhart quasi-polynomials, and a scissors congruence phenomenon

The scissors congruence conjecture for the unimodular group is an analogue of Hilberts 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.