25 research outputs found
Upward planar drawings with two slopes
In an upward planar 2-slope drawing of a digraph, edges are drawn as
straight-line segments in the upward direction without crossings using only two
different slopes. We investigate whether a given upward planar digraph admits
such a drawing and, if so, how to construct it. For the fixed embedding
scenario, we give a simple characterisation and a linear-time construction by
adopting algorithms from orthogonal drawings. For the variable embedding
scenario, we describe a linear-time algorithm for single-source digraphs, a
quartic-time algorithm for series-parallel digraphs, and a fixed-parameter
tractable algorithm for general digraphs. For the latter two classes, we make
use of SPQR-trees and the notion of upward spirality. As an application of this
drawing style, we show how to draw an upward planar phylogenetic network with
two slopes such that all leaves lie on a horizontal line
On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity
Computing planar orthogonal drawings with the minimum number of bends is one
of the most relevant topics in Graph Drawing. The problem is known to be
NP-hard, even when we want to test the existence of a rectilinear planar
drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001).
From the parameterized complexity perspective, the problem is fixed-parameter
tractable when parameterized by the sum of three parameters: the number of
bends, the number of vertices of degree at most two, and the treewidth of the
input graph (Di Giacomo et al., 2022). We improve this last result by showing
that the problem remains fixed-parameter tractable when parameterized only by
the number of vertices of degree at most two plus the number of bends. As a
consequence, rectilinear planarity testing lies in \FPT~parameterized by the
number of vertices of degree at most two.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Rectilinear Planarity of Partial 2-Trees
A graph is rectilinear planar if it admits a planar orthogonal drawing
without bends. While testing rectilinear planarity is NP-hard in general (Garg
and Tamassia, 2001), it is a long-standing open problem to establish a tight
upper bound on its complexity for partial 2-trees, i.e., graphs whose
biconnected components are series-parallel. We describe a new O(n^2)-time
algorithm to test rectilinear planarity of partial 2-trees, which improves over
the current best bound of O(n^3 \log n) (Di Giacomo et al., 2022). Moreover,
for partial 2-trees where no two parallel-components in a biconnected component
share a pole, we are able to achieve optimal O(n)-time complexity. Our
algorithms are based on an extensive study and a deeper understanding of the
notion of orthogonal spirality, introduced several years ago (Di Battista et
al, 1998) to describe how much an orthogonal drawing of a subgraph is rolled-up
in an orthogonal drawing of the graph.Comment: arXiv admin note: substantial text overlap with arXiv:2110.00548
Appears in the Proceedings of the 30th International Symposium on Graph
Drawing and Network Visualization (GD 2022
Optimal Morphs of Planar Orthogonal Drawings
We describe an algorithm that morphs between two planar orthogonal drawings Gamma_I and Gamma_O of a connected graph G, while preserving planarity and orthogonality. Necessarily Gamma_I and Gamma_O share the same combinatorial embedding. Our morph uses a linear number of linear morphs (linear interpolations between two drawings) and preserves linear complexity throughout the process, thereby answering an open question from Biedl et al. [Biedl et al., 2013].
Our algorithm first unifies the two drawings to ensure an equal number of (virtual) bends on each edge. We then interpret bends as vertices which form obstacles for so-called wires: horizontal and vertical lines separating the vertices of Gamma_O. We can find corresponding wires in Gamma_I that share topological properties with the wires in Gamma_O. The structural difference between the two drawings can be captured by the spirality of the wires in Gamma_I, which guides our morph from Gamma_I to Gamma_O
Upward and Orthogonal Planarity are W[1]-hard Parameterized by Treewidth
Upward planarity testing and Rectilinear planarity testing are central
problems in graph drawing. It is known that they are both NP-complete, but XP
when parameterized by treewidth. In this paper we show that these two problems
are W[1]-hard parameterized by treewidth, which answers open problems posed in
two earlier papers. The key step in our proof is an analysis of the
All-or-Nothing Flow problem, a generalization of which was used as an
intermediate step in the NP-completeness proof for both planarity testing
problems. We prove that the flow problem is W[1]-hard parameterized by
treewidth on planar graphs, and that the existing chain of reductions to the
planarity testing problems can be adapted without blowing up the treewidth. Our
reductions also show that the known -time algorithms cannot be
improved to run in time unless ETH fails.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
On Bend-Minimized Orthogonal Drawings of Planar 3-Graphs
An orthogonal drawing of a graph is a planar drawing where each edge is drawn as a sequence of horizontal and vertical line segments. Finding a bend-minimized orthogonal drawing of a planar graph of maximum degree 4 is NP-hard. The problem becomes tractable for planar graphs of maximum degree 3, and the fastest known algorithm takes O(n^5 log n) time. Whether a faster algorithm exists has been a long-standing open problem in graph drawing. In this paper we present an algorithm that takes only O~(n^{17/7}) time, which is a significant improvement over the previous state of the art
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
Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable
The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions