Forbidden Patterns in Mixed Linear Layouts

TL;DR

This paper introduces thick patterns as a new way to characterize mixed linear layouts in ordered graphs, providing a boundedness criterion for graphs with bounded degree and exploring the limits of finite forbidden pattern characterizations.

Contribution

It presents the concept of thick patterns to characterize bounded mixed page numbers in bounded-degree graphs and shows the impossibility of finite forbidden pattern characterizations for fixed mixed page numbers.

Findings

Bounded maximum degree graphs have bounded mixed page number iff largest thick pattern size is bounded.

No finite forbidden pattern characterization exists for fixed mixed page number $k \,\geq\, 2$.

Introduces thick patterns as a new characterization tool for mixed linear layouts.

Abstract

An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue) number of an ordered graph is the minimum number of required stacks (queues). Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. The minimum number of stacks plus queues is called the mixed page number. It is well known that ordered graphs with small stack number are characterized, up to a function, by the absence of large twists (that is, pairwise crossing edges). Similarly, ordered graphs with small queue number are characterized by the absence of large rainbows (that is, pairwise nesting edges). However, no such characterization via forbidden patterns is known for mixed linear layouts. We address this gap by introducing patterns similar to twists and rainbows, which we call thick patterns; such patterns allow a characterization, again up to a function, of mixed linear layouts of bounded-degree graphs. That is, we show that a family of ordered graphs with bounded maximum degree has bounded mixed page number if and only if the size of the largest thick pattern is bounded. In addition, we investigate an exact characterization of ordered graphs whose mixed page number equals a fixed integer $ k $ via a finite set of forbidden patterns. We show that for every $ k \ge 2 $, there is no such characterization, which supports the nature of our first result.