Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs

Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of binary matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any binary matrix that does not have the consecutive-ones property.Comment: A preliminary version of this work appeared in WG13: 39th International Workshop on Graph-Theoretic Concepts in Computer Scienc

On Minimum Saturated Matrices

Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices and the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M contains no submatrix G\in F (as a row and column permutation of G). A matrix M without repeated columns is F-saturated if M is F-admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat(n,F) which is the minimum number of columns of an F-saturated matrix with n rows. We establish the estimate sat(n,F)=O(n^{k-1}) for any family F of k-row matrices and also compute the sat-function for a few small forbidden matrices.Comment: 31 pages, included a C cod

Poly-Bernoulli numbers and lonesum matrices

A lonesum matrix is a matrix that can be uniquely reconstructed from its row and column sums. Kaneko defined the poly-Bernoulli numbers $B_m^{(n)}$ by a generating function, and Brewbaker computed the number of binary lonesum $m\times n$-matrices and showed that this number coincides with the poly-Bernoulli number $B_m^{(-n)}$. We compute the number of $q$-ary lonesum $m\times n$-matrices, and then provide generalized Kaneko's formulas by using the generating function for the number of $q$-ary lonesum $m\times n$-matrices. In addition, we define two types of $q$-ary lonesum matrices that are composed of strong and weak lonesum matrices, and suggest further researches on lonesum matrices. \Comment: 27 page

The highly connected even-cycle and even-cut matroids

The classes of even-cycle matroids, even-cycle matroids with a blocking pair, and even-cut matroids each have hundreds of excluded minors. We show that the number of excluded minors for these classes can be drastically reduced if we consider in each class only the highly connected matroids of sufficient size.Comment: Version 2 is a major revision, including a correction of an error in the statement of one of the main results and improved exposition. It is 89 pages, including a 33-page Jupyter notebook that contains SageMath code and that is also available in the ancillary file