Algorithmic Extensions of Dirac's Theorem

11/06/2020
by   Fedor V. Fomin, et al.
0

In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every n-vertex 2-connected graph G with minimum vertex degree δ≥ 2 contains a cycle with at least min{2δ,n} vertices. In particular, if δ≥ n/2, then G is Hamiltonian. The proof of Dirac's theorem is constructive, and it yields an algorithm computing the corresponding cycle in polynomial time. The combinatorial bound of Dirac's theorem is tight in the following sense. There are 2-connected graphs that do not contain cycles of length more than 2δ+1. Also, there are non-Hamiltonian graphs with all vertices but one of degree at least n/2. This prompts naturally to the following algorithmic questions. For k≥ 1, (A) How difficult is to decide whether a 2-connected graph contains a cycle of length at least min{2δ+k,n}? (B) How difficult is to decide whether a graph G is Hamiltonian, when at least n - k vertices of G are of degrees at least n/2-k? The first question was asked by Fomin, Golovach, Lokshtanov, Panolan, Saurabh, and Zehavi. The second question is due to Jansen, Kozma, and Nederlof. Even for a very special case of k=1, the existence of a polynomial-time algorithm deciding whether G contains a cycle of length at least min{2δ+1,n} was open. We resolve both questions by proving the following algorithmic generalization of Dirac's theorem: If all but k vertices of a 2-connected graph G are of degree at least δ, then deciding whether G has a cycle of length at least min{2δ +k, n} can be done in time 2^𝒪(k)· n^𝒪(1). The proof of the algorithmic generalization of Dirac's theorem builds on new graph-theoretical results that are interesting on their own.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/07/2019

Going Far From Degeneracy

An undirected graph G is d-degenerate if every subgraph of G has a verte...
research
02/07/2022

Longest Cycle above Erdős-Gallai Bound

In 1959, Erdős and Gallai proved that every graph G with average vertex ...
research
02/05/2019

Hamiltonicity below Dirac's condition

Dirac's theorem (1952) is a classical result of graph theory, stating th...
research
05/03/2023

Approximating Long Cycle Above Dirac's Guarantee

Parameterization above (or below) a guarantee is a successful concept in...
research
02/01/2023

Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations

We study the problem of finding a Hamiltonian cycle under the promise th...
research
04/03/2023

Canonical decompositions of 3-connected graphs

We offer a new structural basis for the theory of 3-connected graphs, pr...
research
11/12/2022

Hypercubes and Hamiltonian Cycles of Display Sets of Rooted Phylogenetic Networks

In the context of reconstructing phylogenetic networks from a collection...

Please sign up or login with your details

Forgot password? Click here to reset