Strong cocomparability graphs and Slash-free orderings of matrices

TL;DR

This paper introduces strong cocomparability graphs, characterizes them through forbidden submatrices and orderings, and provides polynomial-time algorithms for recognition, expanding the understanding of matrix rearrangements avoiding specific subpatterns.

Contribution

It defines and characterizes strong cocomparability graphs, providing new algorithms for recognizing these graphs based on matrix permutation avoiding the Slash submatrix.

Findings

Provides a polynomial-time recognition algorithm.

Characterizes the class via forbidden structures.

Connects matrix pattern avoidance to graph classes.

Abstract

We introduce the class of strong cocomparability graphs, as the class of reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows 01, 10, which we call Slash. We provide an ordering characterization, a forbidden structure characterization, and a polynomial-time recognition algorithm, for the class. These results complete the picture in which in addition to, or instead of, the Slash matrix one forbids the Gamma matrix (which has rows 11, 10). It is well known that in these two cases one obtains the class of interval graphs, and the class of strongly chordal graphs, respectively. By complementation, we obtain the class of strong comparability graphs, whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the two-by-two identity submatrix. Thus our results give characterizations and algorithms for this class of irreflexive graphs as well. In other words, our results may be interpreted as solving the following problem: given a symmetric 0,1-matrix with 0-diagonal, can the rows and columns of be simultaneously permuted to avoid the two-by-two identity submatrix?