Layered Separators in Minor-Closed Families with Applications
Vida Dujmović, Pat Morin, David R. Wood · arXiv (Cornell University) · 2013
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.