ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2011, Vol. 5, No. 1, pp. 1–6. c  Pleiades Publishing, Ltd., 2011. Original Russian Text c  V.E. Alekseev, D.V. Zakharova, 2010, published in Diskretnyi Analiz i Issledovanie Operatsii, 2010, Vol. 17, No. 1, pp. 3–10. Independent Sets in the Graphs with Bounded Minors of the Extended Incidence Matrix V. E. Alekseev * and D. V. Zakharova Department of Computational Mathematics and Cybernetics, State University of Nizhnii Novgorod, pr. Gagarina 23, Nizhnii Novgorod, 603950 Russia Received March 2, 2009 Abstract—We characterize the graphs with the absolute values of minors of the extended incidence matrix bounded above by some constant. We prove that, for every fixed k, the independent set problem is solvable in polynomial time for the graphs with the absolute values of minors of the matrix obtained from of the incidence matrix by appending a column of ones at most k. DOI: 10.1134/S1990478911010017 Keywords: extended incidence matrix, minor, independent set problem INTRODUCTION It is known that the integer linear programming problem with a totally unimodular matrix (a matrix all of whose minors take values 0, +1, and −1) is solvable in polynomial time. V. N. Shevchenko suggested [4] that this could be true in the more general case when all minors of the matrix are bounded in absolute value by some constant. There are several versions of this conjecture; two of them involve the extended matrix of the problem: in one case we append the column of the right-hand sides of some inequalities, in the other, the row of coefficients of the target function. We intended to confirm Shevchenko’s conjecture for a particular type of matrices, to wit, the incidence matrix of a graph and two extended versions of it. It is clear below that we are only partially successful. Consider the incidence matrix I = I (G) of a simple graph G (rows correspond to vertices, and columns to edges) and two extended matrices: I ′ results by appending a row of ones; I ′′ results by appending a column of ones. Let M (G), M ′ (G), and M ′′ (G) denote the greatest absolute values of the minors of the matrices I , I ′ , and I ′′ . The first goal of this article is to describe the classes of graphs M k , M ′ k , and M ′′ k with M , M ′ , and M ′′ at most k respectively. All three classes are hereditary in the strong sense; i.e., they are closed under removing vertices and edges. Thus, they can be described by forbidden subgraphs. For the class M k , this description is known [5]: the minimal forbidden subgraphs are the graphs with every connected component being an odd cycle. We show that M ′ k = M k and characterize the forbidden subgraphs for M ′′ k . Also we show that, for every fixed k, the independent set problem for the graphs in M ′′ k is solvable in polynomial time. * E-mail: ave@uic.nnov.ru 1