Recognizing Interval Bigraphs by Forbidden Patterns
  Let H be a connected bipartite graph with n nodes and m edges. We give an O(nm) time algorithm to decide whether H is an interval bigraph. The best known algorithm has time complexity O(nm^6(m + n) \log n) and it was developed in 1997 [18]. Our approach is based on an ordering characterization of interval bigraphs introduced by Hell and Huang [13]. We transform the problem of finding the desired ordering to choosing strong components of a pair-digraph without creating conflicts. We make use of the structure of the pair-digraph as well as decomposition of bigraph H based on the special components of the pair-digraph. This way we make explicit what the difficult cases are and gain efficiency by isolating such situations. 
Arash Rafiey
cs.DS cs.DM math.CO