30 research outputs found
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
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
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
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
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
Orthogonal Graph Drawing with Inflexible Edges
We consider the problem of creating plane orthogonal drawings of 4-planar
graphs (planar graphs with maximum degree 4) with constraints on the number of
bends per edge. More precisely, we have a flexibility function assigning to
each edge a natural number , its flexibility. The problem
FlexDraw asks whether there exists an orthogonal drawing such that each edge
has at most bends. It is known that FlexDraw is NP-hard
if for every edge . On the other hand, FlexDraw can
be solved efficiently if and is trivial if
for every edge .
To close the gap between the NP-hardness for and the
efficient algorithm for , we investigate the
computational complexity of FlexDraw in case only few edges are inflexible
(i.e., have flexibility~). We show that for any FlexDraw
is NP-complete for instances with inflexible edges with
pairwise distance (including the case where they
induce a matching). On the other hand, we give an FPT-algorithm with running
time , where
is the time necessary to compute a maximum flow in a planar flow network with
multiple sources and sinks, and is the number of inflexible edges having at
least one endpoint of degree 4.Comment: 23 pages, 5 figure
Ortho-Radial Drawing in Near-Linear Time
An orthogonal drawing is an embedding of a plane graph into a grid. In a seminal work of Tamassia (SIAM Journal on Computing 1987), a simple combinatorial characterization of angle assignments that can be realized as bend-free orthogonal drawings was established, thereby allowing an orthogonal drawing to be described combinatorially by listing the angles of all corners. The characterization reduces the need to consider certain geometric aspects, such as edge lengths and vertex coordinates, and simplifies the task of graph drawing algorithm design.
Barth, Niedermann, Rutter, and Wolf (SoCG 2017) established an analogous combinatorial characterization for ortho-radial drawings, which are a generalization of orthogonal drawings to cylindrical grids. The proof of the characterization is existential and does not result in an efficient algorithm. Niedermann, Rutter, and Wolf (SoCG 2019) later addressed this issue by developing quadratic-time algorithms for both testing the realizability of a given angle assignment as an ortho-radial drawing without bends and constructing such a drawing.
In this paper, we improve the time complexity of these tasks to near-linear time. We establish a new characterization for ortho-radial drawings based on the concept of a good sequence. Using the new characterization, we design a simple greedy algorithm for constructing ortho-radial drawings