66 research outputs found
RECOGNIZING WEIGHTED AND SEEDED DISK GRAPHS
Disk intersection representations realize graphs by mapping vertices bijectively to disks in the plane such that two disks intersect each other if and only if the corresponding vertices are adjacent in the graph. If intersections are restricted to touching points of the boundaries, we call them disk contact representations. Deciding whether a vertex-weighted planar graph can be realized such that the disks\u27 radii coincide with the vertex weights is known to be NP-hard for both contact and intersection representations. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that in the contact scenario it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present a constructive linear-time recognition algorithm for embedded stars with arbitrary weights.
We also consider a version of the problem in which the disks of a representation are supposed to cover preassigned points, called seeds. We show that both for contact and intersection representations this problem is NP-hard for unit weights even if the given graph is a path. If the disks\u27 radii are not prescribed, the problem remains NP-hard for trees in the contact scenario
An Algorithmic Framework for Labeling Network Maps
Drawing network maps automatically comprises two challenging steps, namely
laying out the map and placing non-overlapping labels. In this paper we tackle
the problem of labeling an already existing network map considering the
application of metro maps. We present a flexible and versatile labeling model.
Despite its simplicity, we prove that it is NP-complete to label a single line
of the network. For a restricted variant of that model, we then introduce an
efficient algorithm that optimally labels a single line with respect to a given
weighting function. Based on that algorithm, we present a general and
sophisticated workflow for multiple metro lines, which is experimentally
evaluated on real-world metro maps.Comment: Full version of COCOON 2015 pape
Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases
Crossing minimization is one of the central problems in graph drawing.
Recently, there has been an increased interest in the problem of minimizing
crossings between paths in drawings of graphs. This is the metro-line crossing
minimization problem (MLCM): Given an embedded graph and a set L of simple
paths, called lines, order the lines on each edge so that the total number of
crossings is minimized. So far, the complexity of MLCM has been an open
problem. In contrast, the problem variant in which line ends must be placed in
outermost position on their edges (MLCM-P) is known to be NP-hard. Our main
results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We
give an -approximation algorithm for MLCM-P
Snapping Graph Drawings to the Grid Optimally
In geographic information systems and in the production of digital maps for
small devices with restricted computational resources one often wants to round
coordinates to a rougher grid. This removes unnecessary detail and reduces
space consumption as well as computation time. This process is called snapping
to the grid and has been investigated thoroughly from a computational-geometry
perspective. In this paper we investigate the same problem for given drawings
of planar graphs under the restriction that their combinatorial embedding must
be kept and edges are drawn straight-line. We show that the problem is NP-hard
for several objectives and provide an integer linear programming formulation.
Given a plane graph G and a positive integer w, our ILP can also be used to
draw G straight-line on a grid of width w and minimum height (if possible).Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Evaluation of Labeling Strategies for Rotating Maps
We consider the following problem of labeling points in a dynamic map that
allows rotation. We are given a set of points in the plane labeled by a set of
mutually disjoint labels, where each label is an axis-aligned rectangle
attached with one corner to its respective point. We require that each label
remains horizontally aligned during the map rotation and our goal is to find a
set of mutually non-overlapping active labels for every rotation angle so that the number of active labels over a full map rotation of
2 is maximized. We discuss and experimentally evaluate several labeling
models that define additional consistency constraints on label activities in
order to reduce flickering effects during monotone map rotation. We introduce
three heuristic algorithms and compare them experimentally to an existing
approximation algorithm and exact solutions obtained from an integer linear
program. Our results show that on the one hand low flickering can be achieved
at the expense of only a small reduction in the objective value, and that on
the other hand the proposed heuristics achieve a high labeling quality
significantly faster than the other methods.Comment: 16 pages, extended version of a SEA 2014 pape
Octilinear Force-Directed Layout with Mental Map Preservation for Schematic Diagrams
We present an algorithm for automatically laying out metro map style schematics using a force-directed approach, where we use a localized version of the standard spring embedder forces combined with an octilinear magnetic force. The two types of forces used during layout are naturally conflicting, and the existing method of simply combining these to generate a resultant force does not give satisfactory results. Hence we vary the forces, emphasizing the standard forces in the beginning to produce a well distributed graph, with the octilinear forces
becoming prevalent at the end of the layout, to ensure that the key requirement of line angles at intervals of 45? is obtained. Our method is considerably faster than the more commonly used search-based approaches, and we believe the results are superior to the previous force-directed approach. We have further developed this technique to address the issues of dynamic schematic layout. We use a Delaunay triangulation to construct a schematic “frame”, which is used to retain relative node positions and permits full control of the level of mental map preservation.
This technique is the first to combine mental map preservation techniques with the additional layout criteria of schematic diagrams. To conclude, we present the results of a study to investigate the relationship between the level of mental map preservation and the user response time and accuracy
Incremental Grid-like Layout Using Soft and Hard Constraints
We explore various techniques to incorporate grid-like layout conventions
into a force-directed, constraint-based graph layout framework. In doing so we
are able to provide high-quality layout---with predominantly axis-aligned
edges---that is more flexible than previous grid-like layout methods and which
can capture layout conventions in notations such as SBGN (Systems Biology
Graphical Notation). Furthermore, the layout is easily able to respect
user-defined constraints and adapt to interaction in online systems and diagram
editors such as Dunnart.Comment: Accepted to Graph Drawing 201
Euclidean Greedy Drawings of Trees
Greedy embedding (or drawing) is a simple and efficient strategy to route
messages in wireless sensor networks. For each source-destination pair of nodes
s, t in a greedy embedding there is always a neighbor u of s that is closer to
t according to some distance metric. The existence of greedy embeddings in the
Euclidean plane R^2 is known for certain graph classes such as 3-connected
planar graphs. We completely characterize the trees that admit a greedy
embedding in R^2. This answers a question by Angelini et al. (Graph Drawing
2009) and is a further step in characterizing the graphs that admit Euclidean
greedy embeddings.Comment: Expanded version of a paper to appear in the 21st European Symposium
on Algorithms (ESA 2013). 24 pages, 20 figure
On the Area Requirements of Planar Greedy Drawings of Triconnected Planar Graphs
In this paper we study the area requirements of planar greedy drawings of
triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family
of subdivisions of triconnected plane graphs and claimed that every planar
greedy drawing of the graphs in respecting the prescribed plane
embedding requires exponential area. However, we show that every -vertex
graph in actually has a planar greedy drawing respecting the
prescribed plane embedding on an grid. This reopens the
question whether triconnected planar graphs admit planar greedy drawings on a
polynomial-size grid. Further, we provide evidence for a positive answer to the
above question by proving that every -vertex Halin graph admits a planar
greedy drawing on an grid. Both such results are obtained by
actually constructing drawings that are convex and angle-monotone. Finally, we
consider -Schnyder drawings, which are angle-monotone and hence greedy
if , and show that there exist planar triangulations for
which every -Schnyder drawing with a fixed requires
exponential area for any resolution rule
Planar Octilinear Drawings with One Bend Per Edge
In octilinear drawings of planar graphs, every edge is drawn as an
alternating sequence of horizontal, vertical and diagonal ()
line-segments. In this paper, we study octilinear drawings of low edge
complexity, i.e., with few bends per edge. A -planar graph is a planar graph
in which each vertex has degree less or equal to . In particular, we prove
that every 4-planar graph admits a planar octilinear drawing with at most one
bend per edge on an integer grid of size . For 5-planar
graphs, we prove that one bend per edge still suffices in order to construct
planar octilinear drawings, but in super-polynomial area. However, for 6-planar
graphs we give a class of graphs whose planar octilinear drawings require at
least two bends per edge
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