Recognition Algorithms for Binary Signed-Graphic Matroids ⋆ Konstantinos Papalamprou 1 and Leonidas Pitsoulis 2 1 Management Science Group, Department of Management, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK 2 Department of Mathematical, Physical and Computational Sciences, Faculty of Engineering, Aristotle University of Thessaloniki,Thessaloniki 54124, Greece Abstract. In this paper we provide two recognition algorithms for the class of signed-graphic matroids along with necessary and sufficient con- ditions for a matroid to be signed-graphic. Specifically, we provide a polynomial-time algorithm which determines whether a given binary ma- troid is signed-graphic and an algorithm which determines whether a gen- eral matroid given by an independence oracle is binary signed-graphic. 1 Introduction Important classes of matrices in combinatorial optimization constitute the real representation matrices for well-known classes of matroids. Most importantly, it was the recognition algorithms of the associated matroids that led to the recognition algorithms for such classes of matrices. Two such celebrated classes are formed by the network and totally unimodular (TU) matrices. In the case of network matrices, it was the recognition algorithm for graphic matroids of Tutte [24] that enabled the first efficient and practical algorithm checking if a matrix is network or not [3] while in the case of TU matrices, it was Seymour’s regular matroid decomposition result [18] that made available the unique recog- nition algorithm for TU matrices. In this paper, we are considering another such important pair, namely: binet matrices and signed-graphic matroids. The main optimization result for binet matrices goes as follows [1,2] if A is a binet matrix then the polyhedron P = {x : Ax ≤ b} has integral vertices for any vector b with even entries (i.e. even vector b). The first recognition algorithm for binet matrices appeared very recently in [12,13]. In our work we utilize this algorithm to provide recognition algorithms for signed-graphic matroids and by this way, we answer an open question posed in [13] concerning the use of that algorithm in recognizing the signed-graphic matroids. However, we believe that the results of this paper along with the binet recognition algorithm will be the ⋆ This research has been funded by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Fund- ing Program: Thalis. Investing in knowledge society through the European Social Fund. A.R. Mahjoub et al. (Eds.): ISCO 2012, LNCS 7422, pp. 463–474, 2012. c  Springer-Verlag Berlin Heidelberg 2012