## 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 ∊ > 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 + ∊)d.

Original language | English |
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Pages (from-to) | 1587-1601 |

Number of pages | 15 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 34 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2020 |

## Keywords

- Above guarantee parameterization
- Fixed-parameter tractability
- Longest cycle
- Longest path

## ASJC Scopus subject areas

- General Mathematics