# Going Far From Degeneracy

## TL;DR

This paper explores the computational complexity of finding long cycles and paths in graphs relative to their degeneracy, revealing polynomial algorithms for certain cases and NP-completeness for others, with implications for graph theory and algorithms.

## Contribution

It establishes fixed-parameter tractable algorithms for detecting long cycles and paths in 2-connected and connected graphs based on degeneracy and introduces optimality results for degeneracy-based parameterization.

## Key findings

- Polynomial-time algorithms for cycles of length at least d+k in 2-connected graphs.
- NP-completeness of deciding long paths in general graphs.
- Optimality of degeneracy as a parameter for long path and cycle problems.

## 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.