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24
Drawing a graph in a hypercube
, 2004
"... A ddimensional hypercube drawing of a graph represents the vertices by distinct points in {0, 1} d, such that the linesegments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to ..."
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A ddimensional hypercube drawing of a graph represents the vertices by distinct points in {0, 1} d, such that the linesegments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections. 1
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.
Plane Drawings of Queue and Deque Graphs ⋆
"... Abstract. In stack and queue layouts the vertices of a graph are linearly ordered from left to right, where each edge corresponds to an item and the left and right end vertex of each edge represents the addition and removal of the item to the used data structure. A graph admitting a stack or queue l ..."
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Abstract. In stack and queue layouts the vertices of a graph are linearly ordered from left to right, where each edge corresponds to an item and the left and right end vertex of each edge represents the addition and removal of the item to the used data structure. A graph admitting a stack or queue layout is a stack or queue graph, respectively. Typical stack and queue layouts are rainbows and twists visualizing the LIFO and FIFO principles, respectively. However, in such visualizations, twists cause many crossings, which make the drawings incomprehensible. We introduce linear cylindric layouts as a visualization technique for queue and deque (doubleended queue) graphs. It provides new insights into the characteristics of these fundamental data structures and extends to the visualization of mixed layouts with stacks and queues. Our main result states that a graph is a deque graph if and only if it has a plane linear cylindric drawing. 1
Boundeddegree graphs have arbitrarily large geometric thickness
 Electron. J. Combin
, 509
"... 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.
Upward threedimensional grid drawings of graphs. arXiv.org math.CO/0510051
, 2005
"... Abstract. A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce threedimensional grid drawings with small bounding box volume. Our first ..."
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Abstract. A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce threedimensional grid drawings with small bounding box volume. Our first main result is that every nvertex graph with bounded degeneracy has a threedimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A threedimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the zdirection. We prove that every dag has an upward threedimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every ccolourable dag (c constant) has an upward threedimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward threedimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward threedimensional grid drawings with bends in the edges. 1.
On the book thickness of ktrees ∗
, 911
"... Every ktree has book thickness at most k + 1, and this bound is best possible for all k ≥ 3. Vandenbussche et al. (2009) proved that every ktree that has a smooth degree3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k ≥ 4, by constructing ..."
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Every ktree has book thickness at most k + 1, and this bound is best possible for all k ≥ 3. Vandenbussche et al. (2009) proved that every ktree that has a smooth degree3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k ≥ 4, by constructing a ktree with book thickness k + 1 that has a smooth degree4 tree decomposition with width k. This solves an open problem of Vandenbussche et al. (2009) 1
unknown title
, 2012
"... We recall that a book with k pages consists of a straight line (the spine) and k halfplanes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is ..."
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We recall that a book with k pages consists of a straight line (the spine) and k halfplanes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a kpage book drawing (or simply a kpage drawing). The pagenumber of a graph G is the minimum k such that G admits a kpage embedding (that is, a kpage drawing with no edge crossings). The kpage crossing number νk(G) of G is the minimum number of crossings in a kpage drawing of G. We investigate the pagenumbers and kpage crossing numbers of complete bipartite graphs. We find the exact pagenumbers of several complete bipartite graphs, and use these pagenumbers to find the exact kpage crossing number of Kk+1,n for k ∈ {3, 4, 5, 6}. We also prove the general asymptotic estimate limk→ ∞ limn→ ∞ νk(Kk+1,n)/(2n2 /k2) = 1. Finally, we give general upper bounds for νk(Km,n), and relate these bounds to the kplanar
1 Graph 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 ..."
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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(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. 1
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.