1,086 research outputs found
Outerplanar graph drawings with few slopes
We consider straight-line outerplanar drawings of outerplanar graphs in which
a small number of distinct edge slopes are used, that is, the segments
representing edges are parallel to a small number of directions. We prove that
edge slopes suffice for every outerplanar graph with maximum degree
. This improves on the previous bound of , which was
shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound
is tight: for every there is an outerplanar graph with maximum
degree that requires at least distinct edge slopes in an
outerplanar straight-line drawing.Comment: Major revision of the whole pape
Stack and Queue Layouts via Layered Separators
It is known that every proper minor-closed class of graphs has bounded
stack-number (a.k.a. book thickness and page number). While this includes
notable graph families such as planar graphs and graphs of bounded genus, many
other graph families are not closed under taking minors. For fixed and ,
we show that every -vertex graph that can be embedded on a surface of genus
with at most crossings per edge has stack-number ;
this includes -planar graphs. The previously best known bound for the
stack-number of these families was , except in the case
of -planar graphs. Analogous results are proved for map graphs that can be
embedded on a surface of fixed genus. None of these families is closed under
taking minors. The main ingredient in the proof of these results is a
construction proving that -vertex graphs that admit constant layered
separators have stack-number.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables
By using some basic calculus of multiple integration, we provide an
alternative expression of the integral in which the minimum and the maximum are replaced
with two single variables. We demonstrate the usefulness of that expression in
the computation of orness and andness average values of certain aggregation
functions. By generalizing our result to Riemann-Stieltjes integrals, we also
provide a method for the calculation of certain expected values and
distribution functions.Comment: 15 page
Aligned Drawings of Planar Graphs
Let be a graph that is topologically embedded in the plane and let
be an arrangement of pseudolines intersecting the drawing of .
An aligned drawing of and is a planar polyline drawing
of with an arrangement of lines so that and are
homeomorphic to and . We show that if is
stretchable and every edge either entirely lies on a pseudoline or it has
at most one intersection with , then and have a
straight-line aligned drawing. In order to prove this result, we strengthen a
result of Da Lozzo et al., and prove that a planar graph and a single
pseudoline have an aligned drawing with a prescribed convex
drawing of the outer face. We also study the less restrictive version of the
alignment problem with respect to one line, where only a set of vertices is
given and we need to determine whether they can be collinear. We show that the
problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th
International Symposium on Graph Drawing and Network Visualization (GD 2017
Mixed Linear Layouts of Planar Graphs
A -stack (respectively, -queue) layout of a graph consists of a total
order of the vertices, and a partition of the edges into sets of
non-crossing (non-nested) edges with respect to the vertex ordering. In 1992,
Heath and Rosenberg conjectured that every planar graph admits a mixed
-stack -queue layout in which every edge is assigned to a stack or to a
queue that use a common vertex ordering.
We disprove this conjecture by providing a planar graph that does not have
such a mixed layout. In addition, we study mixed layouts of graph subdivisions,
and show that every planar graph has a mixed subdivision with one division
vertex per edge.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
Pogrešno i nepotpuno utvrđeno činjenično stanje – pokušaj silovanja
Pogrešno i nepotpuno utvrđeno činjenično stanje –
pokušaj silovanj
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