Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

David Cattanéo 1 Simon Perdrix 2, 3
1 LIG Laboratoire d'Informatique de Grenoble - CAPP
LIG - Laboratoire d'Informatique de Grenoble
3 CARTE - Theoretical adverse computations, and safety
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs.
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Conference papers
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https://hal.archives-ouvertes.fr/hal-01132843
Contributor : Simon Perdrix <>
Submitted on : Wednesday, March 18, 2015 - 10:16:00 AM
Last modification on : Friday, October 25, 2019 - 1:31:03 AM

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David Cattanéo, Simon Perdrix. Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms. 26th International Symposium on Algorithms and Computation (ISAAC 2015), Dec 2015, Nagoya, Japan. pp.12, ⟨10.1007/978-3-662-48971-0_23⟩. ⟨hal-01132843⟩

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