Going far from degeneracy

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.