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Layered Separators in MinorClosed Families with Applications
, 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 minorclos ..."
Abstract

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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 minorclosed 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 queuenumber and nonrepetitive chromatic number of bounded Euler genus graphs. We extend these results, with a log O(1) n bound, to all proper minorclosed families. This result also implies that every graph from a proper minorclosed class has a 3dimensional 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 queuenumber previously known.
Graph Layouts via Layered Separators
"... A kqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of each ver ..."
Abstract

Cited by 3 (2 self)
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A kqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The queuenumber (tracknumber) of a graph G, is the minimum k such that G has a kqueue (ktrack) layout. This paper proves that every nvertex planar graph has track number and queue number at most O(logn). This improves the result of Di Battista, Frati and Pach [Foundations of Computer Science, (FOCS ’10), pp. 365–374] who proved the first subpolynomial bounds on the queue number and track number of planar graphs. Specifically, they obtained O(log 2 n) queue number and O(log 8 n) track number bounds for planar graphs. The result also implies that every planar graph has a 3D crossingfree grid drawing in O(nlogn) volume. The proof uses a nonstandard type of graph separators.
Layered separators for queue layouts, 3D graph drawing and nonrepetitive coloring
, 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 minorclosed ..."
Abstract
 Add to MetaCart
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 minorclosed families. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded 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 layered separators to prove O(log n) bounds for a number of problems where O(√n) was a long standing previous best bound. This includes the nonrepetitive chromatic number and queuenumber of graphs with bounded Euler genus. We extend these results to all proper minorclosed families, with a O(log n) bound on the nonrepetitive chromatic number, and a logO(1) n bound on the queuenumber. Only for planar graphs were logO(1) n bounds previously known. Our results imply that every graph from a proper minorclosed class has a 3dimensional grid drawing with n logO(1) n volume, whereas the previous best bound was O(n3/2). Readers interested in the full details should consult arXiv:1302.0304 and arXiv:1306.1595, rather than the current extended abstract.
1Graph Layouts via Layered Separators
"... Abstract. A kqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A kqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The queuenumber (tracknumber) of a graph G, is the minimum k such that G has a kqueue (ktrack) layout. This paper proves that every nvertex planar graph has track number and queue number at most O(logn). This improves the result of Di Battista, Frati and Pach [Foundations of Computer Science, (FOCS ’10), pp. 365–374] who proved the first subpolynomial bounds on the queue number and track number of planar graphs. Specifically, they obtained O(log2n) queue number and O(log8n) track number bounds for planar graphs. The result also implies that every planar graph has a 3D crossingfree grid drawing in O(n logn) volume. The proof uses a nonstandard type of graph separators. 1