265 research outputs found
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
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
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
Computing NodeTrix Representations of Clustered Graphs
NodeTrix representations are a popular way to visualize clustered graphs;
they represent clusters as adjacency matrices and inter-cluster edges as curves
connecting the matrix boundaries. We study the complexity of constructing
NodeTrix representations focusing on planarity testing problems, and we show
several NP-completeness results and some polynomial-time algorithms. Building
on such algorithms we develop a JavaScript library for NodeTrix representations
aimed at reducing the crossings between edges incident to the same matrix.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Relaxing the Constraints of Clustered Planarity
In a drawing of a clustered graph vertices and edges are drawn as points and
curves, respectively, while clusters are represented by simple closed regions.
A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region,
or region-region crossings. Determining the complexity of testing whether a
clustered graph admits a c-planar drawing is a long-standing open problem in
the Graph Drawing research area. An obvious necessary condition for c-planarity
is the planarity of the graph underlying the clustered graph. However, such a
condition is not sufficient and the consequences on the problem due to the
requirement of not having edge-region and region-region crossings are not yet
fully understood.
In order to shed light on the c-planarity problem, we consider a relaxed
version of it, where some kinds of crossings (either edge-edge, edge-region, or
region-region) are allowed even if the underlying graph is planar. We
investigate the relationships among the minimum number of edge-edge,
edge-region, and region-region crossings for drawings of the same clustered
graph. Also, we consider drawings in which only crossings of one kind are
admitted. In this setting, we prove that drawings with only edge-edge or with
only edge-region crossings always exist, while drawings with only region-region
crossings may not. Further, we provide upper and lower bounds for the number of
such crossings. Finally, we give a polynomial-time algorithm to test whether a
drawing with only region-region crossings exist for biconnected graphs, hence
identifying a first non-trivial necessary condition for c-planarity that can be
tested in polynomial time for a noticeable class of graphs
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
Splitting Clusters To Get C-Planarity
In this paper we introduce a generalization of the c-planarity testing problem for clustered graphs. Namely, given a clustered graph, the goal of the S PLIT-C-P LANARITY problem is to split as few clusters as possible in order to make the graph c-planar. Determining whether zero splits are enough coincides with testing c-planarity. We show that S PLIT-C-P LANARITY is NP-complete for c-connected clustered triangulations and for non-c-connected clustered paths and cycles. On the other hand, we present a polynomial-time algorithm for ïŹat c-connected clustered graphs whose underlying graph is a biconnected seriesparallel graph, both in the ïŹxed and in the variable embedding setting, when the splits are assumed to maintain the c-connectivity of the clusters
Embedding Graphs into Embedded Graphs
A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation.
We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed.
In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings.
To the best of our knowledge an analogous result was previously known essentially only when G is a cycle
Synchronized planarity with applications to constrained planarity problems
We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O(nâž)
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