On Minimally Non-Firm Binary Matrices

TL;DR

This paper investigates the properties of firm and minimally non-firm binary matrices, introducing new concepts and classes, and explores their structural characterizations through graph-theoretic and matrix operations.

Contribution

It introduces the concept of superfirmness, relates it to graph properties, and constructs infinite classes of minimally non-firm matrices using novel matrix operations.

Findings

Superfirmness is equivalent to absence of odd holes in the rectangle cover graph.

Four infinite classes of minimally non-firm matrices are derived.

The work advances towards characterizing firm matrices via forbidden submatrices.

Abstract

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.