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

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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 *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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## References

- 1.M. R. Garey and D. S. Johnson,
*Computers and Intractability*(Freeman, San Fransisco, 1979; Mir, Moscow, 1982).MATHGoogle Scholar - 2.V. E. Alekseev and D. S. Malyshev, “Planar graph classes with the independent set problem solvable in polynomial time,” J. Appl. Ind. Math.
**3**(1), 1–4 (2009).MathSciNetCrossRefGoogle Scholar - 3.A. Biniaz, A. Maheshwari, and M. Smid, “Higher-order triangular-distance Delaunay graphs: Graph-theoretical properties,” Comput. Geometry
**48**(9), 646–660 (2015).MathSciNetCrossRefMATHGoogle Scholar - 4.H. L. Bodlaender, “A partial
*k*-arboretum of graphs with bounded treewidth,” Theoret. Comput. Sci.**209**(1–2), 1–45 (1998).MathSciNetCrossRefMATHGoogle Scholar - 5.C. Brause, N. C. Le, and I. Schiermeyer, “The maximum independent set problem in subclasses of subcubic graphs,” Discrete Math.
**338**(10), 1766–1778 (2015).MathSciNetCrossRefMATHGoogle Scholar - 6.N. Bonichon, C. Gavoille, N. Hanusse, and D. Ilcinkas, “Connections between θ-graphs, Delaunay triangulations, and orthogonal surfaces,” in
*Graph-Theoretic Concepts in Computer Science: Proceedings of the Workshop, Zaros, Crete, 2010*(Springer, Berlin, 2010), Ser. Lecture Notes in Computer Science 6410, pp. 266–278.Google Scholar - 7.G. Das and M. T. Goodrich, “On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees,” Comput. Geometry
**8**(3), 123–137 (1997).MathSciNetCrossRefMATHGoogle Scholar - 8.R. Diestel,
*Graph Theory*(Springer, Berlin, 2010).CrossRefMATHGoogle Scholar - 9.M. B. Dillencourt, “Realizability of Delaunay triangulations,” Inform. Process. Lett.
**33**(6), 283–287 (1990).MathSciNetCrossRefMATHGoogle Scholar - 10.M. B. Dillencourt and W. D. Smith, “Graph-theoretical conditions for inscribability and Delaunay realizability,” Discrete Math.
**161**(1–3), 63–77 (1996).MathSciNetCrossRefMATHGoogle Scholar - 11.G. A. Dirac, “On rigid circuit graphs,” Abh. Math. Sem. Univ. Hamburg
**25**(1), 71–76 (1961).MathSciNetCrossRefMATHGoogle Scholar - 12.H. Fleischner, G. Sabidussi, and I. Sarvanov, “Maximum independent sets in 3-and 4-regular Hamiltonian graphs,” Discrete Math.
**310**(20), 2742–2749 (2010).MathSciNetCrossRefMATHGoogle Scholar - 13.F. Gavril, “Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph,” SIAM J. Comput.
**1**(2), 180–187 (1972).MathSciNetCrossRefMATHGoogle Scholar - 14.M. Kaminski, V. V. Lozin, M. Milanic, “Recent developments on graphs of bounded clique-width,” Discrete Appl. Math.
**157**(12), 2747–2761 (2009).MathSciNetCrossRefMATHGoogle Scholar - 15.D. S. Malyshev, “Classes of subcubic planar graphs for which the independent set problem is polynomially solvable,” J. Appl. Ind. Math.
**7**(4), 537–548 (2013).MathSciNetCrossRefMATHGoogle Scholar - 16.B. Mohar, “Face covers and the genus problem for apex graphs,” J. Combin. Theory, Ser. B
**82**(1), 102–117 (2001).MathSciNetCrossRefMATHGoogle Scholar - 17.G. Narasimhan and M. Smid,
*Geometric Spanner Networks*(Cambridge Univ. Press, Cambridge, 2007).CrossRefMATHGoogle Scholar - 18.P. Bose, V. Dujmovic, F. Hurtado, J. Iacono, S. Langerman, H. Meijer, V. Sacristan, M. Saumell, and D. Wood, “Proximity graphs: E, δ, Δ, χ, and ω,” Internat. J. Comput. Geom. Appl.
**22**(5), 439–470 (2012).MathSciNetCrossRefMATHGoogle Scholar - 19.G. Rote, “Strictly convex drawings of planar graphs,” in
*Proceedings of the 16th Annual ACM–SIAM Symposium on Discrete Algorithms, Vancouver, Canada, 2005*(ACM, New York, 2005), pp. 728–734.Google Scholar - 20.T. K. Dey, M. B. Dillencourt, S. K. Ghosh, and J. M. Cahill, “Triangulating with high connectivity,” Comput. Geometry
**8**(1), 39–56 (1997).MathSciNetCrossRefMATHGoogle Scholar