Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) G, does it contain an induced subdivision of a prescribed digraph D? The complexity of this problem depends on D and on whether H must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs. Key-words: NP-completeness, induced paths and cycles, linkings, 3-SAT ∗ This work was done while the first author was on sabbatical at Team Mascotte, INRIA, Sophia Antipolis France whose hospitality is gratefully acknowledged. Financial support from the Danish National Science research council (FNU) (under grant no. 09-066741) is gratefully acknowledged. † Department of Mathematics and Computer Science, University of Southern Denmark, Odense DK-5230, Denmark (email: [email protected]). ‡ Projet Mascotte, I3S (CNRS, UNSA) and INRIA, Sophia Antipolis. Partly supported by ANR Blanc AGAPE. (email:[email protected]) § CNRS, Université Denis Diderot Paris 7, LIAFA (email: [email protected]) in ria -0 05 27 51 8, v er si on 1 19 O ct 2 01 0 Trouver une subdivision induite d’un digraphe Résumé : Nous considérons le problème suivant: étant donné un graphe orienté (ou un digraphe) G, contient-il une subdivision induite d’un digraphe fixé D? La complexité de ce problème dépend de D et du fait que G ne puisse être qu’un graphe orienté ou puisse contenir des 2-cycles. Nous donnons des exemples d’instances polynomiales ainsi que des preuves de NP-complétude. Mots-clés : NP-complétude, chemins et cycles induits, linkings, 3-SAT in ria -0 05 27 51 8, v er si on 1 19 O ct 2 01 0 Finding an induced subdivision of a digraph 3