999 research outputs found
Advances in C-Planarity Testing of Clustered Graphs
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 c 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 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 this paper, we provide a polynomial time algorithm for c-planarity testing for "almost" c-connected clustered graphs, i.e., graphs for which all c-vertices 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 are connected.
The algorithm uses ideas of the algorithm for subgraph induced planar connectivity augmentation. We regard it as a first step towards general c-planarity testing
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
Clustered Planarity Variants for Level Graphs
We consider variants of the clustered planarity problem for level-planar
drawings. So far, only convex clusters have been studied in this setting. We
introduce two new variants that both insist on a level-planar drawing of the
input graph but relax the requirements on the shape of the clusters. In
unrestricted Clustered Level Planarity (uCLP) we only require that they are
bounded by simple closed curves that enclose exactly the vertices of the
cluster and cross each edge of the graph at most once. The problem y-monotone
Clustered Level Planarity (y-CLP) requires that additionally it must be
possible to augment each cluster with edges that do not cross the cluster
boundaries so that it becomes connected while the graph remains level-planar,
thereby mimicking a classic characterization of clustered planarity in the
level-planar setting.
We give a polynomial-time algorithm for uCLP if the input graph is
biconnected and has a single source. By contrast, we show that y-CLP is hard
under the same restrictions and it remains NP-hard even if the number of levels
is bounded by a constant and there is only a single non-trivial cluster
NodeTrix Planarity Testing with Small Clusters
We study the NodeTrix planarity testing problem for flat clustered graphs
when the maximum size of each cluster is bounded by a constant . We consider
both the case when the sides of the matrices to which the edges are incident
are fixed and the case when they can be chosen arbitrarily. We show that
NodeTrix planarity testing with fixed sides can be solved in
time for every flat clustered graph that can be
reduced to a partial 2-tree by collapsing its clusters into single vertices. In
the general case, NodeTrix planarity testing with fixed sides can be solved in
time for , but it is NP-complete for any . NodeTrix
planarity testing remains NP-complete also in the free sides model when .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
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
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 flat c-connected clustered graphs whose underlying graph is a biconnected seriesparallel graph, both in the fixed and in the variable embedding setting, when the splits are assumed to maintain the c-connectivity of the clusters
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
Triangulating Clustered Graphs
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 mu 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. As we triangulate a planar embedded graph so that G is still planar embedded after triangulation, we consider triangulation of a c -connected clustered graph that preserve the c -planar embedding. In this paper, we provide a linear time algorithm for triangulating c -connected c -planar embedded clustered graphs C=(G,T) so that C is still c -planar embedded after triangulation. We assume that every non-trivial cluster in C has at least two childcluster. This is the first time, this problem was investigated
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