A graph is polar if its vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B 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, cobipartite, and split graphs. The problem of recognizing polar graphs is NP-complete in general. However, it has been shown to be polynomial for several classes of graphs, including cographs and chordal graphs. In this paper, we study the problem of recognizing graphs whose line graphs are polar. It turns out that the core part of this problem lies in determining whether the edge set of a graph admits a partition (A, B) so that A induces a P3-free subgraph (i.e., a matching) and B induces a P4-free subgraph. We give a structural characterization of such graphs. The characterization enables us to devise an O(n) time algorithm to solve the stated recognition problem.
Line-polar graphs: characterization and recognition. Ross Churchley and Jing Huang. SIAM Journal on Discrete Mathematics 25 (3), 1269–1284. August 2011.