dc.contributor.author Fomin, Fedor dc.contributor.author Golovach, Petr dc.contributor.author Lokshtanov, Daniel dc.contributor.author Panolan, Fahad dc.contributor.author Saurabh, Saket dc.contributor.author Zehavi, Meirav dc.date.accessioned 2020-05-13T06:54:34Z dc.date.available 2020-05-13T06:54:34Z dc.date.issued 2019-09-06 dc.Published Fomin V, Golovach P, Lokshtanov D, Panolan F, Saurabh S, Zehavi M. Going Far From Degeneracy. Leibniz International Proceedings in Informatics. 2019; Article No. 47; pp. 47:1–47:14 eng dc.identifier.issn 1868-8969 en_US dc.identifier.uri https://hdl.handle.net/1956/22211 dc.description.abstract An undirected graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. By the classical theorem of Erd\H{o}s and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erd\H{o}s and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least d+1. But can we decide in polynomial time whether a graph contains a cycle of length at least d+2? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: deciding whether a graph has a cycle of length at least d+2 is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether G contains a cycle of length at least d+k can be done in time 2^{O(k)}|V(G)|^{O(1)}. In other words, deciding whether a 2-connected n-vertex G contains a cycle of length at least d+log n can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least d+1 is NP-complete. However, we prove that if graph G is connected, then deciding whether G contains a path of length at least d+k can be done in time 2^{O(k)}n^{O(1)}. We complement these results by showing that the choice of degeneracy as the `above guarantee parameterization' is optimal in the following sense: For any \epsilon>0 it is NP-complete to decide whether a connected (2-connected) graph of degeneracy d has a path (cycle) of length at least (1+\epsilon)d. en_US dc.language.iso eng eng dc.publisher Schloss Dagstuhl en_US dc.rights Attribution CC BY eng dc.rights.uri https://creativecommons.org/licenses/by/3.0/ eng dc.title Going Far From Degeneracy en_US dc.type Peer reviewed dc.type Journal article dc.date.updated 2020-01-17T14:47:07Z dc.description.version publishedVersion en_US dc.rights.holder Copyright © Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi en_US dc.identifier.doi https://doi.org/10.4230/lipics.esa.2019.47 dc.identifier.cristin 1774855 dc.source.journal Leibniz International Proceedings in Informatics dc.relation.project Norges forskningsråd: 263317 dc.relation.project Norges forskningsråd: 249994 dc.identifier.citation Leibniz International Proceedings in Informatics. 2019, 47.
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