关于极小非固定二元矩阵
On Minimally Non-Firm Binary Matrices
摘要：
对于二元矩阵X，布尔秩br（X）是最小整数k，其中X等于k个秩为1的二元矩阵的布尔和，并且隔离数i（X）是其中没有两个在同一行、列和所有1的2x2子矩阵中的1的最大数目。在本文中，我们继续Lubiw的研究企业矩阵。称X是固定的，如果i（X）=br（X），并且这个等式对它的所有子矩阵都成立。我们证明了X的超紧性的强概念等价于X的矩形覆盖图中没有奇洞，在矩形覆盖图中br（X）和i（X）分别转化为团覆盖和独立数.一个二元矩阵是极小非紧致的，如果它不是紧致的，但它的所有真子矩阵都是。我们介绍了两个矩阵操作，导致广义二元矩阵，并使用这些操作，以获得四个无限类的极小非坚定的矩阵。我们希望，我们的工作可能铺平道路走向一个完整的特征，通过禁用子矩阵的公司矩阵。
英文摘要：
For a binary matrix X, the Boolean rank br(X) is the smallest integer k for which X equals the Boolean sum of k rank-1 binary matrices, and the isolation number i(X) is the maximum number of 1s no two of which are in a same row, column and a 2x2 submatrix of all 1s. In this paper, we continue Lubiw's study of firm matrices. X is said to be firm if i(X)=br(X) and this equality holds for all its submatrices. We show that the stronger concept of superfirmness of X is equivalent to having no odd holes in the rectangle cover graph of X, the graph in which br(X) and i(X) translate to the clique cover and the independence number, respectively. A binary matrix is minimally non-firm if it is not firm but all of its proper submatrices are. We introduce two matrix operations that lead to generalised binary matrices and use these operations to derive four infinite classes of minimally non-firm matrices. We hope that our work may pave the way towards a complete characterisation of firm matrices via forbidden submatrices.
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