Upward-Planar Drawings with Bounded Span

2026-05-01

AI Summary

This study addresses the problem of minimizing the maximum edge span in upward planar layered drawings of directed graphs, where edges are required to be y-monotone and non-crossing. The work establishes, for the first time, that this problem is NP-complete even when restricted to directed trees and biconnected single-source digraphs. It further presents fixed-parameter tractable algorithms parameterized by vertex cover number and by the sum of tree depth and span. Through combinatorial graph-theoretic analysis and careful handling of embedding constraints, the paper derives tight upper and lower bounds on the span for directed trees, designs efficient algorithms for specific graph classes such as st-planar graphs, and precisely delineates computational complexity boundaries with respect to several key structural parameters.

Abstract

We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The span of an edge in such a drawing is the absolute difference between the y-coordinates of its endpoints, and the span of the drawing is the maximum span of any edge. The span of an upward-planar graph is the minimum span over all its upward-planar drawings. We study the problem of determining the span of upward-planar graphs and provide both combinatorial and algorithmic results. On the combinatorial side, we present upper and lower bounds for the span of directed trees. On the algorithmic side, we show that the problem of determining the span of an upward-planar graph is NP-complete already for directed trees and for biconnected single-source graphs. Moreover, we give efficient algorithms for several graph families with a bounded number of sources, including st-planar graphs and graphs where the planar or upward-planar embedding is prescribed. Furthermore, we show that the problem is fixed-parameter tractable with respect to the vertex cover number and the treedepth plus the span.

Problem

• upward-planar drawing
• span
• directed graph
• layered drawing
• NP-complete

Innovation

• upward-planar drawing
• span
• NP-completeness
• fixed-parameter tractability
• directed trees