ISAAC 2011: Algorithms and Computation pp 110-119

# Finding Contractions and Induced Minors in Chordal Graphs via Disjoint Paths

• Rémy Belmonte
• Petr A. Golovach
• Pinar Heggernes
• Pim van ’t Hof
• Marcin Kamiński
• Daniël Paulusma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)

## Abstract

The k-Disjoint Paths problem, which takes as input a graph G and k pairs of specified vertices (s i ,t i ), asks whether G contains k mutually vertex-disjoint paths P i such that P i connects s i and t i , for i = 1,…,k. We study a natural variant of this problem, where the vertices of P i must belong to a specified vertex subset U i for i = 1,…,k. In contrast to the original problem, which is polynomial-time solvable for any fixed integer k, we show that this variant is NP-complete even for k = 2. On the positive side, we prove that the problem becomes polynomial-time solvable for any fixed integer k if the input graph is chordal. We use this result to show that, for any fixed graph H, the problems H-Contractibility and H-Induced Minor can be solved in polynomial time on chordal graphs. These problems are to decide whether an input graph G contains H as a contraction or as an induced minor, respectively.

## Keywords

Polynomial Time Maximal Clique Input Graph Disjoint Path Tree Decomposition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Rémy Belmonte
• 1
• Petr A. Golovach
• 2
• Pinar Heggernes
• 1
• Pim van ’t Hof
• 1
• Marcin Kamiński
• 3
• Daniël Paulusma
• 2
1. 1.Department of InformaticsUniversity of BergenNorway
2. 2.School of Engineering and Computing SciencesDurham UniversityUK
3. 3.Département d’InformatiqueUniversité Libre de BruxellesBelgium