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.
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© 1997 Springer-Verlag Berlin Heidelberg
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Prisner, E. (1997). Bicliques in graphs II: Recognizing k-path graphs and underlying graphs of line digraphs. In: Möhring, R.H. (eds) Graph-Theoretic Concepts in Computer Science. WG 1997. Lecture Notes in Computer Science, vol 1335. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024504
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DOI: https://doi.org/10.1007/BFb0024504
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