  Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with
undirected graphs have been defined and studied in several seemingly
independent areas including number theory, discrete geometry, and dynamical
systems. In particular, the authors are motivated by the tropical intersections
problem derived from the Kuramoto equations. Regular subdivisions of adjacency
polytopes are instrumental in solving these problems. This paper explores
connections between the regular subdivisions of an adjacency polytope and the
contraction of the underlying graph along an edge. We construct a special
regular subdivision whose cells are in one-to-one correspondence with facets of
an adjacency polytope associated with an edge-contraction of the original
graph. Moreover, this subdivision induces a decomposition of the original graph
into ``cell subgraphs''. We explore the combinatorial, graph-theoretic, and
matroidal aspects of this connection.
