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# Going Far From Degeneracy

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LIPIcs.ESA.2019.47.pdf
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## Acknowledgements

We thank Nikolay Karpov for communicating to us the question of finding a path above the degeneracy bound and Proposition 15.

## Cite As

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Going Far From Degeneracy. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.47

## 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 Erdős and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erdős 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Longest path
• longest cycle
• fixed-parameter tractability
• above guarantee parameterization

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