Abstract
A set S of k points in the plane is a universal point subset for a class \({\mathcal G}\) of planar graphs if every graph belonging to \({\mathcal G}\) admits a planar straight-line drawing such that k of its vertices are represented by the points of S. In this paper we study the following main problem: For a given class of graphs, what is the maximum k such that there exists a universal point subset of size k? We provide a [\({\sqrt{n} \;}\)] lower bound on k for the class of planar graphs with n vertices. In addition, we consider the value \(F(n, {\mathcal G})\) such that every set of \(F(n, {\mathcal G})\) points in general position is a universal subset for all graphs with n vertices belonging to the family \({\mathcal G}\), and we establish upper and lower bounds for \(F(n, {\mathcal G})\) for different families of planar graphs, including 4-connected planar graphs and nested-triangles graphs.
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Angelini, P. et al. (2012). Universal Point Subsets for Planar Graphs. In: Chao, KM., Hsu, Ts., Lee, DT. (eds) Algorithms and Computation. ISAAC 2012. Lecture Notes in Computer Science, vol 7676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35261-4_45
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DOI: https://doi.org/10.1007/978-3-642-35261-4_45
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