4,603 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
On Upward Drawings of Trees on a Given Grid
Computing a minimum-area planar straight-line drawing of a graph is known to
be NP-hard for planar graphs, even when restricted to outerplanar graphs.
However, the complexity question is open for trees. Only a few hardness results
are known for straight-line drawings of trees under various restrictions such
as edge length or slope constraints. On the other hand, there exist
polynomial-time algorithms for computing minimum-width (resp., minimum-height)
upward drawings of trees, where the height (resp., width) is unbounded.
In this paper we take a major step in understanding the complexity of the
area minimization problem for strictly-upward drawings of trees, which is one
of the most common styles for drawing rooted trees. We prove that given a
rooted tree and a grid, it is NP-hard to decide whether
admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
Notes on large angle crossing graphs
A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges
in G intersect at an angle of at least a. The concept of right angle crossing
(RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown
that any RAC graph with n vertices has at most 4n-10 edges and that there are
infinitely many values of n for which there exists a RAC graph with n vertices
and 4n-10 edges. In this paper, we give upper and lower bounds for the number
of edges in aAC graphs for all 0 < a < Pi/2
Strictly convex drawings of planar graphs
Every three-connected planar graph with n vertices has a drawing on an O(n^2)
x O(n^2) grid in which all faces are strictly convex polygons. These drawings
are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids.
More generally, a strictly convex drawing exists on a grid of size O(W) x
O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds
are obtained when the faces have fewer sides.
In the proof, we derive an explicit lower bound on the number of primitive
vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The
revision includes numerous small additions, corrections, and improvements, in
particular: - a discussion of the constants in the O-notation, after the
statement of thm.1. - a different set-up and clarification of the case
distinction for Lemma
EPG-representations with small grid-size
In an EPG-representation of a graph each vertex is represented by a path
in the rectangular grid, and is an edge in if and only if the paths
representing an share a grid-edge. Requiring paths representing edges
to be x-monotone or, even stronger, both x- and y-monotone gives rise to three
natural variants of EPG-representations, one where edges have no monotonicity
requirements and two with the aforementioned monotonicity requirements. The
focus of this paper is understanding how small a grid can be achieved for such
EPG-representations with respect to various graph parameters.
We show that there are -edge graphs that require a grid of area
in any variant of EPG-representations. Similarly there are
pathwidth- graphs that require height and area in
any variant of EPG-representations. We prove a matching upper bound of
area for all pathwidth- graphs in the strongest model, the one where edges
are required to be both x- and y-monotone. Thus in this strongest model, the
result implies, for example, , and area bounds
for bounded pathwidth graphs, bounded treewidth graphs and all classes of
graphs that exclude a fixed minor, respectively. For the model with no
restrictions on the monotonicity of the edges, stronger results can be achieved
for some graph classes, for example an area bound for bounded treewidth
graphs and bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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