Capturing Lombardi Flow in Orthogonal Drawings by Minimizing the Number of Segments

TL;DR

This paper investigates methods for creating orthogonal graph drawings inspired by Lombardi's artwork, focusing on minimizing segments covering vertices, with polynomial algorithms for specific graph classes.

Contribution

It introduces polynomial algorithms for minimizing segments covering vertices in orthogonal drawings of trees and series-parallel graphs, and proves NP-hardness for the general problem.

Findings

Polynomial-time algorithm for segment minimization in certain graph classes

NP-hardness of the general segment covering problem

Effective algorithms for upward orthogonal drawings of trees and series-parallel graphs

Abstract

Inspired by the artwork of Mark Lombardi, we study the problem of constructing orthogonal drawings where a small number of horizontal and vertical line segments covers all vertices. We study two problems on orthogonal drawings of planar graphs, one that minimizes the total number of line segments and another that minimizes the number of line segments that cover all the vertices. We show that the first problem can be solved by a non-trivial modification of the flow-network orthogonal bend-minimization algorithm of Tamassia, resulting in a polynomial-time algorithm. We show that the second problem is NP-hard even for planar graphs with maximum degree 3. Given this result, we then address this second optimization problem for trees and series-parallel graphs with maximum degree 3. For both graph classes, we give polynomial-time algorithms for upward orthogonal drawings with the minimum number of segments covering the vertices.