Colored Point-set Embeddings of Acyclic Graphs
Emilio Di Giacomo, Leszek Gasieniec, Giuseppe Liotta, Alfredo Navarra

TL;DR
This paper establishes lower bounds on the complexity of planar drawings of certain acyclic graphs with fixed vertex positions, and provides efficient drawing methods for specific graph classes, addressing a long-standing open problem.
Contribution
It proves lower bounds on the number of bends needed in drawings of forests of three stars with fixed vertices, and offers bend-efficient drawings for 3-colored paths and caterpillars.
Findings
Lower bound of Ω(n^{2/3}) edges with Ω(n^{1/3}) bends for certain forests of three stars.
Constant bends per edge achievable for 3-colored paths and caterpillars with leaves of the same color.
Answers a long-standing open problem in graph drawing complexity.
Abstract
We show that any planar drawing of a forest of three stars whose vertices are constrained to be at fixed vertex locations may require edges each having bends in the worst case. The lower bound holds even when the function that maps vertices to points is not a bijection but it is defined by a 3-coloring. In contrast, a constant number of bends per edge can be obtained for 3-colored paths and for 3-colored caterpillars whose leaves all have the same color. Such results answer to a long standing open problem.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Cellular Automata and Applications
