Recognizing line-polar bipartite graphs in time O(n)O(n)

A graph is polar if the vertex set can be partitioned into AA and BB in such a way that AA induces a complete multipartite graph and BB induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete problem in general, and thus it is of interest to restrict the problem to special classes of graphs. Cographs and chordal graphs are among those whose polarity can be recognized in polynomial time. The line-graphs of bipartite graphs are another class of graphs whose polarity has been characterized recently in terms of forbidden subgraphs, but no polynomial time algorithm is given. In this paper, we present an O(n)O(n) algorithm which decides whether the line-graph of an input bipartite graph is polar and constructs a polar partition when one exists.