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

• Torsten Mütze
• Manfred Scheucher
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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