24 research outputs found
On L-shaped point set embeddings of trees : first non-embeddable examples
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∈{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
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Bichromatic Perfect Matchings with Crossings
We consider bichromatic point sets with red and blue points and study
straight-line bichromatic perfect matchings on them. We show that every such
point set in convex position admits a matching with at least
crossings, for some . This bound is tight since for any there exist bichromatic point sets that do not admit
any perfect matching with crossings.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023