182 research outputs found
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
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
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
Planar Embeddings with Small and Uniform Faces
Motivated by finding planar embeddings that lead to drawings with favorable
aesthetics, we study the problems MINMAXFACE and UNIFORMFACES of embedding a
given biconnected multi-graph such that the largest face is as small as
possible and such that all faces have the same size, respectively.
We prove a complexity dichotomy for MINMAXFACE and show that deciding whether
the maximum is at most is polynomial-time solvable for and
NP-complete for . Further, we give a 6-approximation for minimizing
the maximum face in a planar embedding. For UNIFORMFACES, we show that the
problem is NP-complete for odd and even . Moreover, we
characterize the biconnected planar multi-graphs admitting 3- and 4-uniform
embeddings (in a -uniform embedding all faces have size ) and give an
efficient algorithm for testing the existence of a 6-uniform embedding.Comment: 23 pages, 5 figures, extended version of 'Planar Embeddings with
Small and Uniform Faces' (The 25th International Symposium on Algorithms and
Computation, 2014
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