On mixed linear layouts of series-parallel graphs

Abstract

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and 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. In 2017, Pupyrev disproved this conjecture by demonstrating a planar partial 3-tree that does not admit a mixed 1-stack 1-queue layout.In this work, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs. We conclude this by means of a stronger result, stating that the conjecture does not hold, even in the more general union and local settings. In the former, crossings (nestings) are allowed in the stack (queue) as long as the involved edges belong to different connected components of the subgraph induced by its edges, while in the latter several stacks and queues may form the layout, but locally every vertex is incident to edges from only one stack and one queue.