Algorithmic Extensions of Dirac's Theorem

@article{Fomin2022AlgorithmicEO,
  title={Algorithmic Extensions of Dirac's Theorem},
  author={F. Fomin and Petr A. Golovach and Danil Sagunov and Kirill Simonov},
  journal={ArXiv},
  year={2022},
  volume={abs/2011.03619}
}
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 $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$ vertices. In particular, if $\delta\geq n/2$, 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… 
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References

SHOWING 1-10 OF 43 REFERENCES
Hamiltonicity below Dirac's condition
TLDR
The results extend the range of tractability of the Hamiltonian cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound and for the first parameterization it is shown that a kernel with $O(k)$ vertices can be found in polynomial time.
A randomized algorithm for long directed cycle
  • M. Zehavi
  • Mathematics, Computer Science
    Inf. Process. Lett.
  • 2016
TLDR
It is proved that {\sc LDC} can be solved in deterministic time $O^*(\max\{t(G,2k),4^{k+o(k)}\})$ (randomized time O^*(4^k)$).
Determinant Sums for Undirected Hamiltonicity
TLDR
A Monte Carlo algorithm for Hamiltonicity detection in an $n$-vertex undirected graph running in O(1.657^{n}) time is presented, the first superpolynomial improvement on the worst case runtime for the problem since the O*(2^n) bound was established over 50 years ago.
A method in graph theory
TLDR
It is shown that, for many properties P, one can find a suitable value of k such that if C"k(G) has P, then so does G, and this condition for a graph to be hamiltonian is shown to imply the well-known conditions of Chvatal and Las Vergnas.
An Improved Algorithm for Finding Cycles Through Elements
We consider the following problem: Given k independent edges in G. Is there a polynomial time algorithm to decide whether or not G has a cycle through all of these edges ? If the answer is yes,
Generalizations of Dirac's theorem in Hamiltonian graph theory - A survey
  • Hao Li
  • Computer Science, Mathematics
    Discret. Math.
  • 2013
TLDR
A survey on some recent results on generalization of Dirac’s theorem on Hamiltonian graph theory.
Parameterized Traveling Salesman Problem: Beating the Average
TLDR
A considerable generalization of Vizing's result is proved: for each fixed $k$, an algorithm is given that decides whether, for any input edge weighting of $K_n$, there is a Hamilton cycle of weight at most $h(w)-k$ (and constructs such a cycle if it exists).
Long Cycles in Digraphs
Introduction By Ghouila-Houri's theorem [10], every strong digraph of order n and with minimum degree at least n is Hamiltonian. Extensions of this theorem can be found in [11, 13, 16, 18].
Faster deterministic parameterized algorithm for k-Path
  • Dekel Tsur
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 2019
TLDR
A deterministic algorithm for k-Path with time complexity O^*(2.554^k) is given, which improves the previously best deterministic algorithms for this problem of Zehavi [ESA 2015] whose time complexity is 2.597^k.
Finding Detours is Fixed-Parameter Tractable
TLDR
Using insights into structural graph theory, it is proved that the longest detour problem is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n).
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