9 research outputs found
Advancements on SEFE and Partitioned Book Embedding Problems
In this work we investigate the complexity of some problems related to the
{\em Simultaneous Embedding with Fixed Edges} (SEFE) of planar graphs and
the PARTITIONED -PAGE BOOK EMBEDDING (PBE-) problems, which are known to
be equivalent under certain conditions.
While the computational complexity of SEFE for is still a central open
question in Graph Drawing, the problem is NP-complete for [Gassner
{\em et al.}, WG '06], even if the intersection graph is the same for each pair
of graphs ({\em sunflower intersection}) [Schaefer, JGAA (2013)].
We improve on these results by proving that SEFE with and
sunflower intersection is NP-complete even when the intersection graph is a
tree and all the input graphs are biconnected. Also, we prove NP-completeness
for of problem PBE- and of problem PARTITIONED T-COHERENT
-PAGE BOOK EMBEDDING (PTBE-) - that is the generalization of PBE- in
which the ordering of the vertices on the spine is constrained by a tree -
even when two input graphs are biconnected. Further, we provide a linear-time
algorithm for PTBE- when pages are assigned a connected graph.
Finally, we prove that the problem of maximizing the number of edges that are
drawn the same in a SEFE of two graphs is NP-complete in several restricted
settings ({\em optimization version of SEFE}, Open Problem , Chapter of
the Handbook of Graph Drawing and Visualization).Comment: 29 pages, 10 figures, extended version of 'On Some NP-complete SEFE
Problems' (Eighth International Workshop on Algorithms and Computation, 2014
Hierarchical Partial Planarity
In this paper we consider graphs whose edges are associated with a degree of
{\em importance}, which may depend on the type of connections they represent or
on how recently they appeared in the scene, in a streaming setting. The goal is
to construct layouts of these graphs in which the readability of an edge is
proportional to its importance, that is, more important edges have fewer
crossings. We formalize this problem and study the case in which there exist
three different degrees of importance. We give a polynomial-time testing
algorithm when the graph induced by the two most important sets of edges is
biconnected. We also discuss interesting relationships with other
constrained-planarity problems.Comment: Conference version appeared in WG201
Planarity of Streamed Graphs
In this paper we introduce a notion of planarity for graphs that are
presented in a streaming fashion. A is a stream of
edges on a vertex set . A streamed graph is
- with respect to a positive integer window
size if there exists a sequence of planar topological drawings
of the graphs such that
the common graph is drawn the same in
and in , for . The Problem with window size asks whether a given streamed
graph is -stream planar. We also consider a generalization, where there
is an additional whose edges have to be present
during each time step. These problems are related to several well-studied
planarity problems.
We show that the Problem is NP-complete even when
the window size is a constant and that the variant with a backbone graph is
NP-complete for all . On the positive side, we provide
-time algorithms for (i) the case and (ii) all
values of provided the backbone graph consists of one -connected
component plus isolated vertices and no stream edge connects two isolated
vertices. Our results improve on the Hanani-Tutte-style -time
algorithm proposed by Schaefer [GD'14] for .Comment: 21 pages, 9 figures, extended version of "Planarity of Streamed
Graphs" (9th International Conference on Algorithms and Complexity, 2015
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
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
Upward Book Embeddings of st-Graphs
We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k >= 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k=2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth b and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(b)* n+n^3) time and O(n) time, respectively, where f is a singly-exponential function on b. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE