Bicliques in Graphs II: Recognizing k-Path Graphs and Underlying Graphs of Line Digraphs

Given an undirected graph G, Chvatal and Ebenegger showed that deciding whether there is some loopless digraph D such that G is the underlying graph of the line digraph of D is NP-complete. However, we shall show that the question whether there is such a digraph (with loops allowed) with minimum in-and out degrees not less than 2 can be decided in time O(|V|2|E|2) In that case, we show that D is unique modulo reverse, extending previous uniqueness results by Villar.