Track Layouts, Layered Path Decompositions, and Leveled Planarity

TL;DR

This paper explores graph layout parameters like track-number and layered pathwidth, characterizes leveled planar graphs, and examines their computational complexity and relationships with various graph classes.

Contribution

It introduces new characterizations of leveled planar graphs using track layouts and layered path decompositions, and analyzes their computational complexity and parameterized tractability.

Findings

Leveled planar graphs are characterized by track layouts and layered path decompositions.

Determined NP-completeness of track-number and layered pathwidth for leveled planarity.

Identified graph classes with bounded layered pathwidth and analyzed fixed-parameter tractability.

Abstract

We investigate two types of graph layouts, track layouts and layered path decompositions, and the relations between their associated parameters track-number and layered pathwidth. We use these two types of layouts to characterize leveled planar graphs, which are the graphs with planar leveled drawings with no dummy vertices. It follows from the known NP-completeness of leveled planarity that track-number and layered pathwidth are also NP-complete, even for the smallest constant parameter values that make these parameters nontrivial. We prove that the graphs with bounded layered pathwidth include outerplanar graphs, Halin graphs, and squaregraphs, but that (despite having bounded track-number) series-parallel graphs do not have bounded layered pathwidth. Finally, we investigate the parameterized complexity of these layouts, showing that past methods used for book layouts do not work to parameterize the problem by treewidth or almost-tree number but that the problem is (non-uniformly) fixed-parameter tractable for tree-depth.