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Computational Complexity of the Vertex Cover Problem in the Class of Planar Triangulations

  • K. S. Kobylkin
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Abstract

We study the computational complexity of the vertex cover problem in the class of planar graphs (planar triangulations) admitting a plane representation whose faces are triangles. It is shown that the problem is strongly NP-hard in the class of 4-connected planar triangulations in which the degrees of vertices are of order O(log n), where n is the number of vertices, and in the class of plane 4-connected Delaunay triangulations based on the Minkowski triangular distance. A pair of vertices in such a triangulation is adjacent if and only if there is an equilateral triangle ∇(p, λ) with pR2 and λ > 0 whose interior does not contain triangulation vertices and whose boundary contains this pair of vertices and only it, where ∇(p, λ) = p + λ∇ = {xR2: x = p + λa, a ∈ ∇}; here ∇ is the equilateral triangle with unit sides such that its barycenter is the origin and one of the vertices belongs to the negative y-axis. Keywords: computational complexity, Delaunay triangulation, Delaunay TD-triangulation.

Keywords

computational complexity Delaunay triangulation Delaunay TD-triangulation. 

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

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