On L-shaped Point Set Embeddings of Trees: First Non-embeddable Examples
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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∈{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) ≥ n is known to this date. In this paper, we present the first examples of n-vertex trees for n∈{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)≥ n+1 for those n. Moreover, using computer assistance, we show that every tree on n≤ 11 vertices admits an L-shaped embedding in every set of n points, proving that f_d(n)=n, d∈{3,4}, for those n.
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