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On L-Shaped Point Set Embeddings of Trees: First Non-embeddable Examples

  • Torsten Mütze
  • Manfred ScheucherEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

Abstract

An L-shaped embedding of a tree in a point set is a planar drawing of the tree where the vertices are mapped to distinct points of the set and every edge is drawn as a sequence of two axis-aligned line segments. Let \(f_d(n)\) denote the minimum number N of points such that every n-vertex tree with maximum degree \(d\in \{3,4\}\) admits an L-shaped embedding in every point set of size N, where no two points have the same abscissa or ordinate. The best known upper bounds for this problem are \(f_3(n)=O(n^{1.22})\) and \(f_4(n)=O(n^{1.55})\), respectively. However, no lower bound besides the trivial bound \(f_d(n) \ge n\) is known to this date. In this paper, we present the first examples of n-vertex trees for \(n\in \{13,14,16,17,18,19,20\}\) that require strictly more points than vertices to admit an L-shaped embedding, proving that \(f_4(n)\ge n+1\) for those n. Moreover, using computer assistance, we show that every tree on \(n\le 11\) vertices admits an L-shaped embedding in every set of n points, proving that \(f_d(n)=n\), \(d\in \{3,4\}\), for those n.

Keywords

L-shaped embedding Point set Tree SAT 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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