
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 -stack (respectively, -queue) layout of a graph consists of a total order of the vertices, and a partition of the edges into 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 -stack -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.
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