@inproceedings{1fc8117cd0d1493ebdb85d950786d94d,
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 = "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",
}