A criterion of reducibility for a parallelohedron

TL;DR

This paper presents a criterion for determining when a parallelohedron can be decomposed into lower-dimensional parallelohedra, based on the Venkov graph, advancing understanding in discrete geometry.

Contribution

It provides a new criterion for reducibility of parallelohedra using the Venkov graph, refining previous results from the authors' earlier work.

Findings

Established a criterion for reducibility based on the Venkov graph

Connected the reducibility property to graph-theoretic conditions

Extended previous proofs with a revised, simplified argument

Abstract

A parallelohedron is called reducible, if it can be represented as a direct product of two parallelohedra of lower dimension. In his Ph.D. thesis (2005) the first author proved a criterion of reducibility of a parallelohedron in terms of the Venkov graph. In December 2011 the second author presented a slightly revised version of the original proof at the seminar "Discrete Geometry and Geometry of Numbers" (Moscow State University). The present paper follows that talk.