Contracting to a Longest Path in H-Free Graphs

Authors Walter Kern, Daniël Paulusma

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Walter Kern
  • Department of Applied Mathematics, University of Twente, The Netherlands
Daniël Paulusma
  • Department of Computer Science, Durham University, UK

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Walter Kern and Daniël Paulusma. Contracting to a Longest Path in H-Free Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • dichotomy
  • edge contraction
  • path
  • cycle
  • H-free graph


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