An interval bigraph is an undirected bipartite graph whose edge set is the intersection of the edge sets of an interval graph and the edge set of a complete bipartite graph on the same vertex set. A bipartite interval representation of an interval bigraph is given by a bipartitioned set of intervals for its vertices, such that vertices are adjacent if and only if the corresponding intervals intersect and belong to opposite sides of the bipartition. Interval digraphs are directed graphs defined by a closely related concept. Each vertex of an interval digraph is represented by two intervals on the real line, a source interval and a target interval. The directed arc (u, v) exists in the interval digraph if the source interval of u meets the target interval of v. We give a dynamic programming algorithm recognizing interval bigraphs (interval digraphs) in polynomial time. This algorithm recursively constructs a bipartite interval representation of a graph from bipartite interval representations of proper subgraphs. Moreover, we list some forbidden substructures of interval bigraphs.
Müller, H. (1997). Recognizing interval digraphs and interval bigraphs in polynomial time. Discrete Applied Mathematics, 78(1–3), 189–205. https://doi.org/10.1016/S0166-218X(97)00027-9