@inproceedings{729c0b142f294f87a29230785c60b80e,
title = "Going far from degeneracy",
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 2O(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 2O(k)nO(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 ε > 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 + ε)d.",
keywords = "Above guarantee parameterization, Fixed-parameter tractability, Longest cycle, Longest path",
author = "Fomin, {Fedor V.} and Golovach, {Petr A.} and Daniel Lokshtanov and Fahad Panolan and Saket Saurabh and Meirav Zehavi",
note = "Funding Information: The research leading to these results has received funding from the Research Council of Norway via the projects “CLASSIS” and “MULTIVAL” and from the European Research Council (ERC) via grant LOPPRE, reference 819416. Funding Information: Funding The research leading to these results has received funding from the Research Council of Norway via the projects “CLASSIS” and “MULTIVAL” and from the European Research Council (ERC) via grant LOPPRE, reference 819416. Publisher Copyright: {\textcopyright} Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi.; 27th Annual European Symposium on Algorithms, ESA 2019 ; Conference date: 09-09-2019 Through 11-09-2019",
year = "2019",
month = sep,
day = "1",
doi = "10.4230/LIPIcs.ESA.2019.47",
language = "English",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",
editor = "Bender, {Michael A.} and Ola Svensson and Grzegorz Herman",
booktitle = "27th Annual European Symposium on Algorithms, ESA 2019",
address = "Germany",
}