Colored Point-set Embeddings of Acyclic Graphs

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 $\Omega(n^\frac{2}{3})$ edges each having $\Omega(n^\frac{1}{3})$ 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.