eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
22:1
22:18
10.4230/LIPIcs.ISAAC.2020.22
article
Contracting to a Longest Path in H-Free Graphs
Kern, Walter
1
Paulusma, Daniël
2
https://orcid.org/0000-0001-5945-9287
Department of Applied Mathematics, University of Twente, The Netherlands
Department of Computer Science, Durham University, UK
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol181-isaac2020/LIPIcs.ISAAC.2020.22/LIPIcs.ISAAC.2020.22.pdf
dichotomy
edge contraction
path
cycle
H-free graph