14 research outputs found
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
Track Layouts of Graphs
A \emph{-track layout} of a graph consists of a (proper) vertex
-colouring of , a total order of each vertex colour class, and a
(non-proper) edge -colouring such that between each pair of colour classes
no two monochromatic edges cross. This structure has recently arisen in the
study of three-dimensional graph drawings. This paper presents the beginnings
of a theory of track layouts. First we determine the maximum number of edges in
a -track layout, and show how to colour the edges given fixed linear
orderings of the vertex colour classes. We then describe methods for the
manipulation of track layouts. For example, we show how to decrease the number
of edge colours in a track layout at the expense of increasing the number of
tracks, and vice versa. We then study the relationship between track layouts
and other models of graph layout, namely stack and queue layouts, and geometric
thickness. One of our principle results is that the queue-number and
track-number of a graph are tied, in the sense that one is bounded by a
function of the other. As corollaries we prove that acyclic chromatic number is
bounded by both queue-number and stack-number. Finally we consider track
layouts of planar graphs. While it is an open problem whether planar graphs
have bounded track-number, we prove bounds on the track-number of outerplanar
graphs, and give the best known lower bound on the track-number of planar
graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as
Technical Report TR-2003-07, School of Computer Science, Carleton University,
Ottawa, Canad
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
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
On the queue number of planar graphs
We prove that planar graphs have O(log2 n) queue number, thus improving upon the previous O(Formula Presented) upper bound. Consequently, planar graphs admit three-dimensional straight-line crossing-free grid drawings in O(n log8 n) volume, thus improving upon the previous O(n3/2) upper bound. © 2013 Society for Industrial and Applied Mathematics
Visualizing three-dimensional graph drawings
viii, 110 leaves : ill. (some col.) ; 29 cm.The GLuskap system for interactive three-dimensional graph drawing applies techniques of
scientific visualization and interactive systems to the construction, display, and analysis of
graph drawings. Important features of the system include support for large-screen stereographic
3D display with immersive head-tracking and motion-tracked interactive 3D wand
control. A distributed rendering architecture contributes to the portability of the system,
with user control performed on a laptop computer without specialized graphics hardware.
An interface for implementing graph drawing layout and analysis algorithms in the Python
programming language is also provided. This thesis describes comprehensively the work
on the system by the author—this work includes the design and implementation of the major
features described above. Further directions for continued development and research in
cognitive tools for graph drawing research are also suggested