On opposition graphs, coalition graphs, and bipartite permutation graphs

A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs still remain long-standing open problems. The present paper gives characterizations for co-bipartite opposition graphs and co-bipartite coalition graphs, and for bipartite opposition graphs. Implicit in our argument is a linear time recognition algorithm for these graphs. As an interesting by-product, we find new submatrix characterizations for the well-studied bipartite permutation graphs.