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.
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