Strong Cocomparability Graphs and Slash-Free Orderings of Matrices

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 828-844, March 2024.
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 certifying recognition algorithm for the class. These results complete the picture in which in addition to, or instead of, the [math] matrix one forbids the [math] 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?