57 research outputs found
Layout of Graphs with Bounded Tree-Width
A \emph{queue layout} of a graph consists of a total order of the vertices,
and a partition of the edges into \emph{queues}, such that no two edges in the
same queue are nested. The minimum number of queues in a queue layout of a
graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line
grid) drawing} of a graph represents the vertices by points in
and the edges by non-crossing line-segments. This paper contributes three main
results:
(1) It is proved that the minimum volume of a certain type of
three-dimensional drawing of a graph is closely related to the queue-number
of . In particular, if is an -vertex member of a proper minor-closed
family of graphs (such as a planar graph), then has a drawing if and only if has O(1) queue-number.
(2) It is proved that queue-number is bounded by tree-width, thus resolving
an open problem due to Ganley and Heath (2001), and disproving a conjecture of
Pemmaraju (1992). This result provides renewed hope for the positive resolution
of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional
drawings with O(n) volume. This is the most general family of graphs known to
admit three-dimensional drawings with O(n) volume.
The proofs depend upon our results regarding \emph{track layouts} and
\emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October
2002. This paper incorporates the following conference papers: (1) Dujmovic',
Morin & Wood. Path-width and three-dimensional straight-line grid drawings of
graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts,
tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS
2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of
-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200
On obstacle numbers
The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala et al. (2012) show that there exist graphs with n vertices having obstacle number in Ω(n/ log n). In this note, we up this lower bound to Ω(n/(log log n)2). Our proof makes use of an upper bound of Mukkamala et al. on the number of graphs having obstacle number at most h in such a way that any subsequent improvements to their upper bound will improve our lower bound
Upward Three-Dimensional Grid Drawings of Graphs
A \emph{three-dimensional 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
three-dimensional grid drawings with small bounding box volume. We prove that
every -vertex graph with bounded degeneracy has a three-dimensional grid
drawing with volume. This is the broadest class of graphs admiting
such drawings. A three-dimensional grid drawing of a directed graph is
\emph{upward} if every arc points up in the z-direction. We prove that every
directed acyclic graph has an upward three-dimensional grid drawing with
volume, which is tight for the complete dag. The previous best upper
bound was . Our main result is that every -colourable directed
acyclic graph ( constant) has an upward three-dimensional grid drawing with
volume. This result matches the bound in the undirected case, and
improves the best known bound from for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar
A linear-time algorithm for finding a complete graph minor in a dense graph
Let g(t) be the minimum number such that every graph G with average degree
d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as
originally shown by Mader. Kostochka and Thomason independently proved that
g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon
> 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq
(2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This
improves a previous result by Reed and Wood who gave a linear-time algorithm
when d(G) \geq 2^{t-2}.Comment: 6 pages, 0 figures; Clarification added in several places, no change
to arguments or result
Crossings in grid drawings
We prove tight crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of crossing-free geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m ≥ 8N edges and with vertices on a 3D integer grid of volume N, has Ω((m2/N) log(m/N)) crossings. In d-dimensions, with d ≥ 4, this bound becomes Ω(m2/N). We provide matching upper bounds for all d. Finally, for d ≥ 4 the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some d-dimensional grid of volume N is NΘ(N). In 3 dimensions it remains open to improve the trivial bounds, namely, the 2Ω(N) lower bound and the NO(N) upper bound
An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains
It is known that the First-Fit algorithm for partitioning a poset P into
chains uses relatively few chains when P does not have two incomparable chains
each of size k. In particular, if P has width w then Bosek, Krawczyk, and
Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound
of ckw^{2} on the number of chains used by First-Fit for some constant c, while
Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this
paper we prove an upper bound of the form ckw. This is best possible up to the
value of c.Comment: v3: referees' comments incorporate
Nonrepetitive colourings of planar graphs with O(log n) colours
A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive k-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is O(n−−√) for n-vertex planar graphs. We prove a O(logn) upper bound
Every Large Point Set contains Many Collinear Points or an Empty Pentagon
We prove the following generalised empty pentagon theorem: for every integer
, every sufficiently large set of points in the plane contains
collinear points or an empty pentagon. As an application, we settle the
next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and
Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]
Nonrepetitive Colouring via Entropy Compression
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively -choosable if given lists of at least colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
is -choosable, for some constant . We prove this result
with (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth is nonrepetitively -colourable.Comment: v4: Minor changes made following helpful comments by the referee
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in ℝ2, there is a line ℓ such that in both line sets, for both halfplanes delimited by ℓ, there are √n lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are √n/3 of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdo{double acute}s-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to graph drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labeled lines that are universal for all n-vertex labeled planar graphs. In contrast, the main result by Pach and Toth (J. Graph Theory 46(1):39-47, 2004), has, as an easy consequence, that every set of n (unlabeled) lines is universal for all n-vertex (unlabeled) planar graphs. © 2012 Springer Science+Business Media, LLC.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
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