Stack and Queue Layouts via Layered Separators

TL;DR

This paper proves that certain non-minor-closed graph families, such as $k$-planar and map graphs on fixed-genus surfaces, have stack-number logarithmic in the number of vertices, improving previous bounds.

Contribution

It establishes that graphs with constant layered separators on fixed-genus surfaces have logarithmic stack-number, extending known results beyond minor-closed classes.

Findings

Graphs on fixed-genus surfaces with limited crossings per edge have $oxed{ ext{O}( ext{log } n)}$ stack-number.

New bounds for stack-number of $k$-planar and map graphs on fixed-genus surfaces.

Main technique involves constructing graphs with constant layered separators leading to logarithmic stack-number.

Abstract

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed $g$ and $k$, we show that every $n$-vertex graph that can be embedded on a surface of genus $g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$; this includes $k$-planar graphs. The previously best known bound for the stack-number of these families was $\mathcal{O}(\sqrt{n})$, except in the case of $1$-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that $n$-vertex graphs that admit constant layered separators have $\mathcal{O}(\log n)$ stack-number.