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 and every edge is drawn as a sequence of two axis-aligned line segments. There has been considerable work on establishing upper bounds on the minimum cardinality of a point set to guarantee that any tree of the same size with maximum degree 4 admits an L-shaped embedding on the point set.
However, no non-trivial lower bound is known to this date, i.e., no known n-vertex tree requires more than n points to be embedded. In this paper, we present the first examples of n-vertex trees for n ELEMENT OF {13, 14, 16, 17, 18, 19, 20} that require strictly more points than vertices to admit an L-shaped embedding.
Moreover, using computer help, we show that every tree on n <= 12 vertices admits an L-shaped embedding in every set of n points. We also consider embedding ordered trees, where the cyclic order of the neighbors of each vertex in the embedding is prescribed.
For this setting, we determine the smallest non-embeddable ordered tree on n = 10 vertices, and we show that every ordered tree on n <= 9 or n = 11 vertices admits an L-shaped embedding in every set of n points. We also construct an infinite family of ordered trees which do not always admit an L-shaped embedding, answering a question raised by Biedl, Chan, Derka, Jain, and Lubiw.