15 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
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
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
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