Results 11 
19 of
19
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
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. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract
 Add to MetaCart
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 ..."
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 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.
The 27th Workshop on Combinatorial Mathematics and Computation Theory Queue layouts on folded hypercubes F Qn with n � 7
"... A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn(G), is called the queuenumber of G. An ndimensional folded h ..."
Abstract
 Add to MetaCart
A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn(G), is called the queuenumber of G. An ndimensional folded hypercube, denoted by F Qn, is an enhanced ndimensional hypercube with one extra edge between vertices that have the furthest Hamming distance. In this paper, we deal with queue layout of folded hypercubes and contribute some results as follows: (1) qn(F Qn) = 2 if n ∈ {2, 3}. (2) 2 � qn(F Q4) � 4. (3) 2 � qn(F Q5) � 6. (4) 2 � qn(F Q6) � 7. (5) 3 � qn(F Q7) � 12.
Queue layouts of hypercubes ∗
, 2011
"... A queue layout of a graph consists of a linear ordering σ of its vertices, and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to σ. We show that the ndimensional hypercube Qn has a layout into n−⌊log2 n⌋ queues for all n ≥ 1. On the oth ..."
Abstract
 Add to MetaCart
A queue layout of a graph consists of a linear ordering σ of its vertices, and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to σ. We show that the ndimensional hypercube Qn has a layout into n−⌊log2 n⌋ queues for all n ≥ 1. On the other hand, for every ε> 0 every queue layout of Qn has more than ( 1 2 − ε)n − O(1/ε) queues, and in particular, more than (n − 2)/3 queues. This improves previously known upper and lower bounds on the minimal number of queues in a queue layout of Qn. For the lower bound we employ a new technique of outin representations and contractions which may be of independent interest.