Graph edge contraction and subdivisions for adjacency polytopes

TL;DR

This paper investigates how contracting edges in a graph relates to the regular subdivisions of adjacency polytopes, revealing a correspondence that aids in understanding complex geometric and combinatorial structures.

Contribution

It establishes a novel link between graph edge contractions and regular subdivisions of adjacency polytopes, including a construction of subdivisions with cells corresponding to contracted graph facets.

Findings

Constructed a regular subdivision linked to edge contraction.

Showed cells of subdivision correspond to facets of contracted graph polytope.

Explored combinatorial and matroidal properties of the subdivision.

Abstract

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.