Core Concepts

The authors provide an algorithmic generalization of Dirac's theorem, showing that for a 2-connected graph G, deciding whether G contains a cycle of length at least min{2δ(G-B), |V(G)|-|B|} + k can be done in time 2^O(k+|B|) * n^O(1), where B is a subset of vertices and k is an integer.

Abstract

The paper presents an algorithmic generalization of Dirac's theorem on long cycles in graphs with large minimum vertex degrees. The key contributions are:
The authors introduce a new graph decomposition called Dirac decomposition, which is useful for algorithmically enlarging cycles in graphs. This decomposition has properties that enable efficient algorithms.
They provide an algorithm to solve the Long Dirac Cycle problem, which asks to decide whether a 2-connected graph G contains a cycle of length at least min{2δ(G-B), |V(G)|-|B|} + k, where B is a subset of vertices and k is an integer. The algorithm runs in time 2^O(k+|B|) * n^O(1).
As auxiliary results, the authors also solve the Long (s,t)-Cycle and Long Erd??s-Gallai (s,t)-Path problems, which are of independent interest.
They resolve the conjecture of Jansen, Kozma, and Nederlof on the parameterized complexity of deciding Hamiltonicity when at least n-k vertices have degree at least n/2-k.
The paper combines new graph-theoretic insights with advanced algorithmic techniques to obtain the main result, which significantly generalizes and extends Dirac's classical theorem from the algorithmic perspective.

Stats

Every n-vertex 2-connected graph G with minimum vertex degree δ ≥2 contains a cycle with at least min{2δ, n} vertices.
If δ ≥n/2, then G is Hamiltonian.

Quotes

"Every n-vertex 2-connected graph G with minimum vertex degree δ ≥2 contains a cycle with at least min{2δ, n} vertices."
"If δ ≥n/2, then G is Hamiltonian."

Key Insights Distilled From

by Fedor V. Fom... at **arxiv.org** 04-10-2024

Deeper Inquiries

The techniques developed in the paper, such as Dirac decomposition and Erd˝os-Gallai decomposition, can be applied to various other graph problems beyond long cycles and paths. These decomposition methods provide a structured way to analyze the connectivity and relationships within a graph. For example, they can be utilized in problems related to network flow optimization, graph coloring, graph connectivity, and even in designing efficient routing algorithms in communication networks. By understanding the structural properties of graphs through these decomposition techniques, researchers can develop algorithms for a wide range of graph-related challenges.

The Dirac decomposition approach, while powerful in providing insights into the structure of graphs, has its limitations. One limitation is that it heavily relies on the 2-connectivity of the graph, which restricts its applicability to graphs that are not 2-connected. Additionally, the complexity of the decomposition process may increase significantly for larger graphs or graphs with complex connectivity patterns. To further generalize or extend the Dirac decomposition approach, researchers could explore variations that are applicable to a broader range of graph types, such as directed graphs or weighted graphs. Developing more efficient algorithms for the decomposition process could also enhance its usability in practical applications.

The single-exponential dependence on the parameters k and |B| in the running time of the algorithms developed in the paper is optimal under standard complexity-theoretic assumptions. This is in line with the Fixed-Parameter Tractability (FPT) framework, where the running time of an algorithm is expressed as a function of the parameter(s) and the input size. The single-exponential dependence signifies that the algorithms are efficient for moderate values of the parameters but may become computationally expensive for large parameter values. Improving the running time beyond single-exponential dependence would likely require breakthroughs in algorithm design or new complexity-theoretic insights.

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