[Paper Review] Layered Separators in Minor-Closed Families with Applications

Vida Dujmović, Pat Morin+1 more|arXiv (Cornell University)|Jun 7, 2013

Advanced Graph Theory Research70 references16 citations

TL;DR

This paper introduces layered separators—graph separators with bounded breadth despite potentially linear size in n—for minor-closed families, including bounded Euler genus graphs. By leveraging these separators, the authors achieve O(log n) bounds for queue-number and nonrepetitive chromatic number, improving over the longstanding O(√n) bound, and extend this to O(log^O(1) n) for all proper minor-closed families, enabling improved 3D grid drawings with n log^O(1) n volume.

ABSTRACT

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω ( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed families. We study a special type of graph separator, called a layered separator, which possibly has linear size in n, but has constant size with respect to a different measure, called the breadth. We prove that a wide class of graphs admit layered separators of bounded breadth, including graphs of bounded Euler genus. We use these results to prove O(log n) bounds for a number of problems where O ( √ n) was a long standing previous best bound. This includes queue-number and nonrepetitive chromatic number of bounded Euler genus graphs. We extend these results, with a log O(1) n bound, to all proper minor-closed families. This result also implies that every graph from a proper minor-closed class has a 3-dimensional grid drawing in n log O(1) n volume, where the previous best bound was O(n 3/2). Only for planar graphs was a log O(1) n bound on the queue-number previously known.

Motivation & Objective

• To overcome the limitation of standard separators being as large as Ω(√n) in minor-closed families, such as planar graphs.
• To develop a new separator type—layered separators—that maintains constant size with respect to a new measure, breadth, even when linear in n.
• To establish O(log n) bounds for key graph parameters like queue-number and nonrepetitive chromatic number in bounded Euler genus graphs.
• To extend these results to all proper minor-closed families using layered separators.
• To apply layered separators to improve 3D grid drawing volume bounds from O(n^{3/2}) to n log^{O(1)} n.

Proposed method

• Define layered separators as separators with bounded breadth, where breadth measures the number of layers in a layered graph decomposition.
• Prove that graphs of bounded Euler genus admit layered separators of bounded breadth.
• Use layered separators to bound the queue-number and nonrepetitive chromatic number via divide-and-conquer and dynamic coloring techniques.
• Extend the framework to all proper minor-closed families using structural graph theory and minor-excluded decomposition.
• Apply the separator framework to 3D grid drawings by embedding graphs using layered separators to minimize volume.
• Leverage known results on minor-closed families to show that the log^O(1) n volume bound holds universally across such classes.

Experimental results

Research questions

• RQ1 Can layered separators with bounded breadth be constructed in minor-closed families, particularly those of bounded Euler genus?
• RQ2 What is the impact of layered separators on improving bounds for queue-number and nonrepetitive chromatic number?
• RQ3 Can the O(log n) bound for queue-number be extended beyond planar graphs to all proper minor-closed families?
• RQ4 What is the minimum volume required for 3D grid drawings of graphs in minor-closed families using layered separators?
• RQ5 How does the breadth of a layered separator influence the asymptotic complexity of graph parameters?

Key findings

• Graphs of bounded Euler genus admit layered separators of bounded breadth, enabling improved bounds on key graph parameters.
• The queue-number and nonrepetitive chromatic number of bounded Euler genus graphs are bounded by O(log n), improving over the previous O(√n) bound.
• These O(log n) bounds extend to all proper minor-closed families, achieving O(log^O(1) n) complexity for the same parameters.
• Every graph in a proper minor-closed family has a 3D grid drawing in volume n log^{O(1)} n, improving the prior O(n^{3/2}) bound.
• The result confirms that planar graphs are not unique in achieving log^O(1) n volume for 3D grid drawings—this extends to all minor-closed families.
• Layered separators provide a unifying framework to achieve improved asymptotic bounds across multiple graph problems in minor-closed families.