Mixed Linear Layouts of Planar Graphs

TL;DR

This paper disproves a longstanding conjecture by showing that not all planar graphs admit a mixed 1-stack 1-queue layout, and explores mixed layouts of graph subdivisions, revealing new structural insights.

Contribution

It disproves the conjecture that all planar graphs have a mixed 1-stack 1-queue layout and studies mixed layouts of subdivisions, establishing new limitations and possibilities.

Findings

A specific planar graph without a mixed 1-stack 1-queue layout.

Every planar graph has a mixed subdivision with one division vertex per edge.

Disproof of Heath and Rosenberg's conjecture from 1992.

Abstract

A $k$-stack (respectively, $k$-queue) layout of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing (non-nested) edges with respect to the vertex ordering. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed $1$-stack $1$-queue layout in which every edge is assigned to a stack or to a queue that use a common vertex ordering. We disprove this conjecture by providing a planar graph that does not have such a mixed layout. In addition, we study mixed layouts of graph subdivisions, and show that every planar graph has a mixed subdivision with one division vertex per edge.