  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?
