Master polytopes for cycles of binary matroids

Abstract

Prior work on the cycle polytopes P( M) of binary matroids M has almost exclusively concentrated on regular matroids. Yet almost all binary matroids are nonregular, and almost nothing is known about their cycle polytopes. In this paper we introduce a class of binary matroids L k , k⩾1, the complete binary matroids of order k. We show that the facets of the cycle polytopes P( L k ) have a rather simple description which may be used to deduce easily some, and in principle all, facets of the cycle polytopes of general binary matroids M. For this reason we call the polytopes P( L k ) master polytopes. Specifically, we describe two methods by which facets of P( M) can be deduced from the facets of certain master polytopes. One method produces a complete description of P( M) but is not computationally efficient. The other one produces a subset of the facets of P( M) by an efficient lifting procedure.