164 research outputs found
A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem
The clustered planarity problem (c-planarity) asks whether a hierarchically
clustered graph admits a planar drawing such that the clusters can be nicely
represented by regions. We introduce the cd-tree data structure and give a new
characterization of c-planarity. It leads to efficient algorithms for
c-planarity testing in the following cases. (i) Every cluster and every
co-cluster (complement of a cluster) has at most two connected components. (ii)
Every cluster has at most five outgoing edges.
Moreover, the cd-tree reveals interesting connections between c-planarity and
planarity with constraints on the order of edges around vertices. On one hand,
this gives rise to a bunch of new open problems related to c-planarity, on the
other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure
Advances on Testing C-Planarity of Embedded Flat Clustered Graphs
We show a polynomial-time algorithm for testing c-planarity of embedded flat
clustered graphs with at most two vertices per cluster on each face.Comment: Accepted at GD '1
Detecting Weakly Simple Polygons
A closed curve in the plane is weakly simple if it is the limit (in the
Fr\'echet metric) of a sequence of simple closed curves. We describe an
algorithm to determine whether a closed walk of length n in a simple plane
graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time
algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary,
we obtain the first efficient algorithm to determine whether an arbitrary
n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We
also describe algorithms that detect weak simplicity in O(n log n) time for two
interesting classes of polygons. Finally, we discuss subtle errors in several
previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201
Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity
The C-Planarity problem asks for a drawing of a ,
i.e., a graph whose vertices belong to properly nested clusters, in which each
cluster is represented by a simple closed region with no edge-edge crossings,
no region-region crossings, and no unnecessary edge-region crossings. We study
C-Planarity for , graphs with a fixed
combinatorial embedding whose clusters partition the vertex set. Our main
result is a subexponential-time algorithm to test C-Planarity for these graphs
when their face size is bounded. Furthermore, we consider a variation of the
notion of in which, for each face,
including the outer face, there is a bag that contains every vertex of the
face. We show that C-Planarity is fixed-parameter tractable with the
embedded-width of the underlying graph and the number of disconnected clusters
as parameters.Comment: 14 pages, 6 figure
C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width
For a clustered graph, i.e, a graph whose vertex set is recursively
partitioned into clusters, the C-Planarity Testing problem asks whether it is
possible to find a planar embedding of the graph and a representation of each
cluster as a region homeomorphic to a closed disk such that 1. the subgraph
induced by each cluster is drawn in the interior of the corresponding disk, 2.
each edge intersects any disk at most once, and 3. the nesting between clusters
is reflected by the representation, i.e., child clusters are properly contained
in their parent cluster. The computational complexity of this problem, whose
study has been central to the theory of graph visualization since its
introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades.
Planarity for clustered graphs. ESA'95], has only been recently settled
[Radoslav Fulek and Csaba D. T\'oth. Atomic Embeddability, Clustered Planarity,
and Thickenability. To appear at SODA'20]. Before such a breakthrough, the
complexity question was still unsolved even when the graph has a prescribed
planar embedding, i.e, for embedded clustered graphs.
We show that the C-Planarity Testing problem admits a single-exponential
single-parameter FPT algorithm for embedded clustered graphs, when
parameterized by the carving-width of the dual graph of the input. This is the
first FPT algorithm for this long-standing open problem with respect to a
single notable graph-width parameter. Moreover, in the general case, the
polynomial dependency of our FPT algorithm is smaller than the one of the
algorithm by Fulek and T\'oth. To further strengthen the relevance of this
result, we show that the C-Planarity Testing problem retains its computational
complexity when parameterized by several other graph-width parameters, which
may potentially lead to faster algorithms.Comment: Extended version of the paper "C-Planarity Testing of Embedded
Clustered Graphs with Bounded Dual Carving-Width" to appear in the
Proceedings of the 14th International Symposium on Parameterized and Exact
Computation (IPEC 2019
Constrained Planarity in Practice -- Engineering the Synchronized Planarity Algorithm
In the constrained planarity setting, we ask whether a graph admits a planar
drawing that additionally satisfies a given set of constraints. These
constraints are often derived from very natural problems; prominent examples
are Level Planarity, where vertices have to lie on given horizontal lines
indicating a hierarchy, and Clustered Planarity, where we additionally draw the
boundaries of clusters which recursively group the vertices in a crossing-free
manner. Despite receiving significant amount of attention and substantial
theoretical progress on these problems, only very few of the found solutions
have been put into practice and evaluated experimentally.
In this paper, we describe our implementation of the recent quadratic-time
algorithm by Bl\"asius et al. [TALG Vol 19, No 4] for solving the problem
Synchronized Planarity, which can be seen as a common generalization of several
constrained planarity problems, including the aforementioned ones. Our
experimental evaluation on an existing benchmark set shows that even our
baseline implementation outperforms all competitors by at least an order of
magnitude. We systematically investigate the degrees of freedom in the
implementation of the Synchronized Planarity algorithm for larger instances and
propose several modifications that further improve the performance. Altogether,
this allows us to solve instances with up to 100 vertices in milliseconds and
instances with up to 100 000 vertices within a few minutes.Comment: to appear in Proceedings of ALENEX 202
Constrained Planarity and Augmentation Problems
A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E). Each vertex m in T corresponds to a subset of the vertices of the graph called ``cluster''. c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown by Dahlhaus, Eades, Feng, Cohen that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In the first part of the thesis, we provide a polynomial time algorithms for c-planarity testing of specific planar clustered graphs: Graphs for which - all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each non-connected cluster its super-cluster and all its siblings in T are connected, - for all clusters m G-G(m) is connected. The algorithms are based on the concepts for the subgraph induced planar connectivity augmentation problem, also presented in this thesis. Furthermore, we give some characterizations of c-planar clustered graphs using minors and dual graphs and introduce a c-planar augmentation method. Parts II deals with edge deletion and bimodal crossing minimization. We prove that the maximum planar subgraph problem remains NP-complete even for non-planar graphs without a minor isomorphic to either K(5) or K(3,3), respectively. Further, we investigate the problem of finding a minimum weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Finally, we investigate the problem of drawing a directed graph in two dimensions with a minimal number of crossings such that for every node the incoming and outgoing edges are separated consecutively in the cyclic adjacency lists. It turns out that the planarization method can be adapted such that the number of crossings can be expected to grow only slightly for practical instances
New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity
Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity
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