Bicliques in graphs II: Recognizing k-path graphs and underlying graphs of line digraphs

Abstract

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.

The k-path graph P _k (H) of a graph H has all length-k paths of H as vertices; two such vertices are adjacent in the new graph if their union forms a path or cycle of length k + 1 in H, and if the edge-intersection of both paths forms a path of length k − 1. We also show that, given a graph G = (V, E), there is an O(|V|⁴)-time algorithm that decides whether there is some graph H of minimum degree at least k + 1 with G = P _k(H). If it is, we show that k and H are unique, extending previous uniqueness results by Xueliang Li.

The algorithms are rather similiar and work with the bicliques—inclusion-maximal induced complete bipartite subgraphs—of the graphs. Cruical is the fact that underlying graphs of line digraphs, as well as k-path graphs contain only ‘few’ large bicliques (i.e. bicliques containing K _2,2).

Mathematisches Seminar, Universität Hamburg Bundesstr. 55, 20146 Hamburg, Germany; supported by the Deutsche Forschungsgemeinschaft under grant no. Pr 324/61; part of this research was done at Clemson University, whose hospitality is greatly aknowledged.