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31
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 ..."
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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.
Layouts of Graph Subdivisions
, 2004
"... A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of e ..."
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A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges 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 stacknumber (respectively, queuenumber, tracknumber) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a kstack (kqueue, ktrack) layout. This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3stack subdivision. The best known upper bound on the number of division vertices per edge in a 3stack subdivision of an nvertex graph G is improved from O(log n) to O(log min{sn(G), qn(G)}). This result reduces the question of whether queuenumber is bounded by stacknumber to whether 3stack graphs have bounded queue number. It is proved that every graph has a 2queue subdivision, a 4track subdivision, and a mixed 1stack 1queue subdivision. All these values are optimal for every nonplanar graph. In addition, we characterise those graphs with kstack, kqueue, and ktrack subdivisions, for all values of k. The number of division vertices per edge in the case of 2queue and 4track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge.
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 ..."
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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.
Boundeddegree graphs have arbitrarily large queuenumber
, 2008
"... It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c. ..."
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It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c.
Vertex partitions of chordal graphs
 J. Graph Theory
"... Abstract. A ktree is a chordal graph with no (k + 2)clique. An ℓtreepartition of a graph G is a vertex partition of G into ‘bags’, such that contracting each bag to a single vertex gives an ℓtree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ℓ ≥ ..."
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Abstract. A ktree is a chordal graph with no (k + 2)clique. An ℓtreepartition of a graph G is a vertex partition of G into ‘bags’, such that contracting each bag to a single vertex gives an ℓtree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ℓ ≥ 0, every ktree has an ℓtreepartition in which every bag induces a connected ⌊k/(ℓ + 1)⌋tree. An analogous result is proved for oriented ktrees. 1.
Characterizations and Examples of Graph Classes with bounded expansion
"... Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of th ..."
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Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several lineartime algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of socalled topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with ErdösRényi model of random graphs with constant average degree. In particular, we prove that for every fixed d >
Really straight drawings II: Nonplanar graphs
, 2005
"... We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings ..."
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We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.
Nothreeinlinein3D
 In Proc. 12th Int. Symp. on Graph Drawing (GD’04) [GD004
, 2004
"... The nothreeinline problem, introduced by Dudeney in 1917, asks for the maximum number of points in the nn grid with no three points collinear. In 1951, Erdos proved that the answer is (n). We consider the analogous threedimensional problem, and prove that the maximum number of points in the ..."
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The nothreeinline problem, introduced by Dudeney in 1917, asks for the maximum number of points in the nn grid with no three points collinear. In 1951, Erdos proved that the answer is (n). We consider the analogous threedimensional problem, and prove that the maximum number of points in the n n n grid with no three collinear is (n ).