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 [Formula: see text] matrix one forbids the [Formula: see text] 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?

We unify two popular graph classes, strongly chordal graphs and chordal bigraphs, by introducing an umbrella class that contains both classes and maintains their essential properties. This is done by allowing loops at vertices. Considering loops often has little impact on a class of graphs; it however makes a big difference in this case. We call the new class \itstrongly chordal graphs with possible loops. When all vertices have loops, we recover the usual strongly chordal graphs; when all vertices are loopless, we obtain the usual chordal bigraphs. Moreover, there is a surprizing wealth of graphs in the new class that have loops at some vertices and not at others. These graphs also admit the elegant algorithms previously only applied in the extreme two cases. Formulated in the language of adjacency matrices, we study the class of symmetric 0, 1 matrices that admit a simultaneous row and column permutation avoiding the $\Gamma$ matrix $[ \begin{smallmatrix} 1 \ 1 \\ 1 \ 0 \end{smallmatrix}]$. We give ordering characterizations, matrix characterizations, and forbidden subgraph characterizations of the new class, and illustrate its usefulness by solving the minimum domination problem in this general context. This implies solutions of both the minimum dominating set in strongly chordal graphs and the minimum total dominating set in chordal bigraphs.

We propose bipartite analogues of comparability and cocomparability graphs. Surprisingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderings, forbidden substructures, and orientations of their complements. In particular, we prove that cocomparability bigraphs are precisely those bipartite graphs that do not have edge-asteroids; this is analogous to Gallai's structural characterization of cocomparability graphs by the absence of (vertex-) asteroids. Our characterizations imply a robust polynomial-time recognition algorithm for the class of cocomparability bigraphs. Finally, we also discuss a natural relation of cocomparability bigraphs to interval containment bigraphs, resembling a well-known relation of cocomparability graphs to interval graphs.