81 research outputs found
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
Computing k-Modal Embeddings of Planar Digraphs
Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.
First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [https://doi.org/10.1007/978-3-319-73915-1_37], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [https://doi.org/10.7155/jgaa.00437], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [https://doi.org/10.1007/978-3-319-73915-1_36], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. [https://doi.org/10.1007/978-3-540-24605-3_14]. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family
Approximation Algorithms for Facial Cycles in Planar Embeddings
Consider the following combinatorial problem: Given a planar graph G and a set of simple cycles C in G, find a planar embedding E of G such that the number of cycles in C that bound a face in E is maximized. This problem, called Max Facial C-Cycles, was first studied by Mutzel and Weiskircher [IPCO \u2799, http://dx.doi.org/10.1007/3-540-48777-8_27) and then proved NP-hard by Woeginger [Oper. Res. Lett., 2002, http://dx.doi.org/10.1016/S0167-6377(02)00119-0].
We establish a tight border of tractability for Max Facial C-Cycles in biconnected planar graphs by giving conditions under which the problem is NP-hard and showing that strengthening any of these conditions makes the problem polynomial-time solvable. Our main results are approximation algorithms for Max Facial C-Cycles. Namely, we give a 2-approximation for series-parallel graphs and a (4+epsilon)-approximation for biconnected planar graphs. Remarkably, this provides one of the first approximation algorithms for constrained embedding problems
Clustered Planarity with Pipes
We study the version of the C-Planarity problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the Strip Planarity problem. We give algorithms to decide several families of instances for the two variants in which the order of the pipes around each cluster is given as part of the input or can be chosen by the algorithm
Strip Planarity Testing of Embedded Planar Graphs
In this paper we introduce and study the strip planarity testing problem,
which takes as an input a planar graph and a function and asks whether a planar drawing of exists
such that each edge is monotone in the -direction and, for any
with , it holds . The problem has strong
relationships with some of the most deeply studied variants of the planarity
testing problem, such as clustered planarity, upward planarity, and level
planarity. We show that the problem is polynomial-time solvable if has a
fixed planar embedding.Comment: 24 pages, 12 figures, extended version of 'Strip Planarity Testing'
(21st International Symposium on Graph Drawing, 2013
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
Graph Product Structure for h-Framed Graphs
Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmovi? et al., J. ACM, 67(4), 22:1-38, 2020].
In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size 3? h/2 ?+? h/3 ?-1. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 115 to 82 and from ? 33/2(k+3 ? k/2? -3)? to ? 33/2 (3? k/2 ?+? k/3 ?-1) ?, respectively. We also employ the product structure machinery to improve the current upper bounds on the twin-width of 1-planar graphs from O(1) to 80. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions
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