1,219 research outputs found
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
New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity
Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity
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
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
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
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