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

## 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 *p* ∈ *R*^{2} 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∇ = {*x* ∈ *R*^{2}: x = *p* + λ*a*, *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.## Preview

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