On Mixed Linear Layouts of Series-Parallel Graphs

TL;DR

This paper investigates the limitations of mixed linear layouts in series-parallel graphs, demonstrating that even simple subclasses like 2-trees cannot always be represented with a 1-stack 1-queue layout, challenging previous conjectures.

Contribution

It proves that the conjecture that all planar graphs admit a 1-stack 1-queue layout does not hold for series-parallel graphs, specifically 2-trees.

Findings

Not all series-parallel graphs admit a 1-stack 1-queue layout.

The conjecture by Heath and Rosenberg is false for 2-trees.

Mixed linear layout limitations extend beyond planar graphs.

Abstract

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pupyrev disproved this conjectured by demonstrating a planar partial 3-tree that does not admit a 1-stack 1-queue layout. In this note, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs.