102 research outputs found
Upward and Orthogonal Planarity are W[1]-hard Parameterized by Treewidth
Upward planarity testing and Rectilinear planarity testing are central
problems in graph drawing. It is known that they are both NP-complete, but XP
when parameterized by treewidth. In this paper we show that these two problems
are W[1]-hard parameterized by treewidth, which answers open problems posed in
two earlier papers. The key step in our proof is an analysis of the
All-or-Nothing Flow problem, a generalization of which was used as an
intermediate step in the NP-completeness proof for both planarity testing
problems. We prove that the flow problem is W[1]-hard parameterized by
treewidth on planar graphs, and that the existing chain of reductions to the
planarity testing problems can be adapted without blowing up the treewidth. Our
reductions also show that the known -time algorithms cannot be
improved to run in time unless ETH fails.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Graph Planarity Testing with Hierarchical Embedding Constraints
Hierarchical embedding constraints define a set of allowed cyclic orders for
the edges incident to the vertices of a graph. These constraints are expressed
in terms of FPQ-trees. FPQ-trees are a variant of PQ-trees that includes
F-nodes in addition to P- and to Q-nodes. An F-node represents a permutation
that is fixed, i.e., it cannot be reversed. Let be a graph such that every
vertex of is equipped with a set of FPQ-trees encoding hierarchical
embedding constraints for its incident edges. We study the problem of testing
whether admits a planar embedding such that, for each vertex of ,
the cyclic order of the edges incident to is described by at least one of
the FPQ-trees associated with~. We prove that the problem is fixed-parameter
tractable for biconnected graphs, where the parameters are the treewidth of
and the number of FPQ-trees associated with every vertex of . We also show
that the problem is NP-complete if parameterized by the number of FPQ-trees
only, and W[1]-hard if parameterized by the treewidth only. Besides being
interesting on its own right, the study of planarity testing with hierarchical
embedding constraints can be used to address other planarity testing problems.
In particular, we apply our techniques to the study of NodeTrix planarity
testing of clustered graphs. We show that NodeTrix planarity testing with fixed
sides is fixed-parameter tractable when parameterized by the size of the
clusters and by the treewidth of the multi-graph obtained by collapsing the
clusters to single vertices, provided that this graph is biconnected
Multilevel Planarity
In this paper, we introduce and study multilevel planarity, a generalization of upward planarity and level planarity. Let be a directed graph and let be a function that assigns a finite set of integers to each vertex. A multilevel-planar drawing of is a planar drawing of such that for each vertex its -coordinate is in , nd each edge is drawn as a strictly -monotone curve. We present linear-time algorithms for testing multilevel planarity of embedded graphs with a single source and of oriented cycles. Complementing these algorithmic results, we show that multilevel-planarity testing is NP-complete even in very restricted cases
Sketched Representations and Orthogonal Planarity of Bounded Treewidth Graphs
Given a planar graph and an integer , OrthogonalPlanarity is the
problem of deciding whether admits an orthogonal drawing with at most
bends in total. We show that OrthogonalPlanarity can be solved in polynomial
time if has bounded treewidth. Our proof is based on an FPT algorithm whose
parameters are the number of bends, the treewidth and the number of degree-2
vertices of . This result is based on the concept of sketched orthogonal
representation that synthetically describes a family of equivalent orthogonal
representations. Our approach can be extended to related problems such as
HV-Planarity and FlexDraw. In particular, both OrthogonalPlanarity and
HV-Planarity can be decided in time for series-parallel graphs,
which improves over the previously known bounds.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
Embedding a graph in the grid of a surface with the minimum number of bends is NP-hard
This paper is devoted to the study of graph embeddings in the grid of non-planar surfaces. We provide an adequate model for those embeddings and we study the complexity of minimizing the number of bends. In particular, we prove that testing whether a graph admits a rectilinear (without bends) embedding essentially equivalent to a given embedding, and that given a graph, testing if there exists a surface such that the graph admits a rectilinear embedding in that surface are NP-complete problems and hence the corresponding optimization problems are NP-hard
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
Upward Planar Morphs
We prove that, given two topologically-equivalent upward planar straight-line
drawings of an -vertex directed graph , there always exists a morph
between them such that all the intermediate drawings of the morph are upward
planar and straight-line. Such a morph consists of morphing steps if
is a reduced planar -graph, morphing steps if is a planar
-graph, morphing steps if is a reduced upward planar graph, and
morphing steps if is a general upward planar graph. Further, we
show that morphing steps might be necessary for an upward planar
morph between two topologically-equivalent upward planar straight-line drawings
of an -vertex path.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018) The current version is the
extended on
Drawing graphs for cartographic applications
Graph Drawing is a relatively young area that combines elements of graph theory, algorithms, (computational) geometry and (computational) topology. Research in this field concentrates on developing algorithms for drawing graphs while satisfying certain aesthetic criteria. These criteria are often expressed in properties like edge complexity, number of edge crossings, angular resolutions, shapes of faces or graph symmetries and in general aim at creating a drawing of a graph that conveys the information to the reader in the best possible way. Graph drawing has applications in a wide variety of areas which include cartography, VLSI design and information visualization. In this thesis we consider several graph drawing problems. The first problem we address is rectilinear cartogram construction. A cartogram, also known as value-by-area map, is a technique used by cartographers to visualize statistical data over a set of geographical regions like countries, states or counties. The regions of a cartogram are deformed such that the area of a region corresponds to a particular geographic variable. The shapes of the regions depend on the type of cartogram. We consider rectilinear cartograms of constant complexity, that is cartograms where each region is a rectilinear polygon with a constant number of vertices. Whether a cartogram is good is determined by how closely the cartogram resembles the original map and how precisely the area of its regions describe the associated values. The cartographic error is defined for each region as jAc¡Asj=As, where Ac is the area of the region in the cartogram and As is the specified area of that region, given by the geographic variable to be shown. In this thesis we consider the construction of rectilinear cartograms that have correct adjacencies of the regions and zero cartographic error. We show that any plane triangulated graph admits a rectilinear cartogram where every region has at most 40 vertices which can be constructed in O(nlogn) time. We also present experimental results that show that in practice the algorithm works significantly better than suggested by the complexity bounds. In our experiments on real-world data we were always able to construct a cartogram where the average number of vertices per region does not exceed five. Since a rectangle has four vertices, this means that most of the regions of our rectilinear car tograms are in fact rectangles. Moreover, the maximum number vertices of each region in these cartograms never exceeded ten. The second problem we address in this thesis concerns cased drawings of graphs. The vertices of a drawing are commonly marked with a disk, but differentiating between vertices and edge crossings in a dense graph can still be difficult. Edge casing is a wellknown method—used, for example, in electrical drawings, when depicting knots, and, more generally, in information visualization—to alleviate this problem and to improve the readability of a drawing. A cased drawing orders the edges of each crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. One can also envision that every edge is encased in a strip of the background color and that the casing of the upper edge covers the lower edge at the crossing. If there are no application-specific restrictions that dictate the order of the edges at each crossing, then we can in principle choose freely how to arrange them. However, certain orders will lead to a more readable drawing than others. In this thesis we formulate aesthetic criteria for a cased drawing as optimization problems and solve these problems. For most of the problems we present either a polynomial time algorithm or demonstrate that the problem is NP-hard. Finally we consider a combinatorial question in computational topology concerning three types of objects: closed curves in the plane, surfaces immersed in the plane, and surfaces embedded in space. In particular, we study casings of closed curves in the plane to decide whether these curves can be embedded as the boundaries of certain special surfaces. We show that it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we can determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n=2 combinatorially distinct spatial embeddings and we discuss the existence of fixed-parameter tractable algorithms for related problems
Drawing HV-Restricted Planar Graphs
A strict orthogonal drawing of a graph in is a
drawing of such that each vertex is mapped to a distinct point and each
edge is mapped to a horizontal or vertical line segment. A graph is
-restricted if each of its edges is assigned a horizontal or vertical
orientation. A strict orthogonal drawing of an -restricted graph is
good if it is planar and respects the edge orientations of . In this paper,
we give a polynomial-time algorithm to check whether a given -restricted
plane graph (i.e., a planar graph with a fixed combinatorial embedding) admits
a good orthogonal drawing preserving the input embedding, which settles an open
question posed by Ma\v{n}uch et al. (Graph Drawing 2010). We then examine
-restricted planar graphs (i.e., when the embedding is not fixed), and give
a complete characterization of the -restricted biconnected outerplanar
graphs that admit good orthogonal drawings.Comment: 17 pages, 9 figure
Radial level planarity with fixed embedding
We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions of combinatorial embeddings, namely the level embedding, the upward embedding and the planar embedding. These notions allow for increasing degrees of freedom in their corresponding drawings. For the fixed level embedding there are known and easy to test level planarity criteria. We use these criteria to prove an "untangling" lemma that plays a key role in a simple level planarity test for the case where only the upward embedding is fixed. This test is then adapted to the case where only the planar embedding is fixed. Further, we characterize radial upward planar embeddings, which lets us extend our results to radial level planarity. No algorithms were previously known for these problems
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