Subdivisions of digraphs

Ana Karolinna Maia De Oliveira 1
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : In this work, we consider the following problem: Given a directed graph D, does it contain a subdivision of a prescribed digraph F? We believe that there is a dichotomy between NP-complete and polynomial-time solvable instances of this problem. We present many examples of both cases. In particular, except for five instances, we are able to classify all the digraphs F of order 4.While all NP-hardness proofs are made by reduction from some version of the 2-linkage problem in digraphs, we use different algorithmic tools for proving polynomial-time solvability of certain instances, some of them involving relatively complicated algorithms. The techniques vary from easy brute force algorithms, algorithms based on maximum-flow calculations, handle decompositions of strongly connected digraphs, among others. Finally, we treat the very special case of F being the disjoint union of directed cycles. In particular, we show that the directed cycles of length at least 3 have the Erdos-Pósa Property: for every n, there exists an integer tn such that for every digraph D, either D contains n disjoint directed cycles of length at least 3, or there is a set T of tn vertices that meets every directed cycle of length at least 3. From this result, we deduce that if F is the disjoint union of directed cycles of length at most 3, then one can decide in polynomial time if a digraph contains a subdivision of F.
Document type :
Complete list of metadatas
Contributor : Abes Star <>
Submitted on : Friday, March 6, 2015 - 11:27:58 PM
Last modification on : Monday, November 5, 2018 - 3:36:03 PM
Long-term archiving on : Sunday, June 7, 2015 - 8:50:55 PM


Version validated by the jury (STAR)


  • HAL Id : tel-01127012, version 1



Ana Karolinna Maia De Oliveira. Subdivisions of digraphs. Other [cs.OH]. Université Nice Sophia Antipolis, 2014. English. ⟨NNT : 2014NICE4084⟩. ⟨tel-01127012⟩



Record views


Files downloads