127 research outputs found
Relating Graph Thickness to Planar Layers and Bend Complexity
The thickness of a graph with vertices is the minimum number of
planar subgraphs of whose union is . A polyline drawing of in
is a drawing of , where each vertex is mapped to a
point and each edge is mapped to a polygonal chain. Bend and layer complexities
are two important aesthetics of such a drawing. The bend complexity of
is the maximum number of bends per edge in , and the layer complexity
of is the minimum integer such that the set of polygonal chains in
can be partitioned into disjoint sets, where each set corresponds
to a planar polyline drawing. Let be a graph of thickness . By
F\'{a}ry's theorem, if , then can be drawn on a single layer with bend
complexity . A few extensions to higher thickness are known, e.g., if
(resp., ), then can be drawn on layers with bend complexity 2
(resp., ). However, allowing a higher number of layers may reduce the
bend complexity, e.g., complete graphs require layers to be drawn
using 0 bends per edge.
In this paper we present an elegant extension of F\'{a}ry's theorem to draw
graphs of thickness . We first prove that thickness- graphs can be
drawn on layers with bends per edge. We then develop another
technique to draw thickness- graphs on layers with bend complexity,
i.e., , where . Previously, the bend complexity was not known to be sublinear for
. Finally, we show that graphs with linear arboricity can be drawn on
layers with bend complexity .Comment: A preliminary version appeared at the 43rd International Colloquium
on Automata, Languages and Programming (ICALP 2016
Experimental Evaluation of Book Drawing Algorithms
A -page book drawing of a graph consists of a linear ordering of
its vertices along a spine and an assignment of each edge to one of the
pages, which are half-planes bounded by the spine. In a book drawing, two edges
cross if and only if they are assigned to the same page and their vertices
alternate along the spine. Crossing minimization in a -page book drawing is
NP-hard, yet book drawings have multiple applications in visualization and
beyond. Therefore several heuristic book drawing algorithms exist, but there is
no broader comparative study on their relative performance. In this paper, we
propose a comprehensive benchmark set of challenging graph classes for book
drawing algorithms and provide an extensive experimental study of the
performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
A Linear Time Algorithm for Drawing a Graph in 3 Pages within its Isotopy Class in 3-Space
We consider undirected graphs up to an ambient isotopy in 3-space. Such a graph can be represented by a plane diagram or a Gauss code. We recognize in linear time if a Gauss code represents an actual graph in 3-space. We also design a linear time algorithm for drawing a topological 3-page embedding of a graph isotopic to a given graph
A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages
We introduce simple codes and fast visualization tools for knotted structures in molecules and neural networks. Knots, links and more general knotted graphs are studied up to an ambient isotopy in Euclidean 3-space. A knotted graph can be represented by a plane diagram or by an abstract Gauss code. First we recognize in linear time if an abstract Gauss code represents an actual graph embedded in 3-space. Second we design a fast algorithm for drawing any knotted graph in the 3-page book, which is a union of 3 half-planes along their common boundary line. The running time of our drawing algorithm is linear in the length of a Gauss code of a given graph. Three-page embeddings provide simple linear codes of knotted graphs so that the isotopy problem for all graphs in 3-space completely reduces to a word problem in finitely presented semigroups
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
Characterisations and Examples of Graph Classes with Bounded Expansion
Classes with bounded expansion, which generalise classes that exclude a
topological minor, have recently been introduced by Ne\v{s}et\v{r}il and Ossona
de Mendez. These classes are defined by the fact that the maximum average
degree of a shallow minor of a graph in the class is bounded by a function of
the depth of the shallow minor. Several linear-time algorithms are known for
bounded expansion classes (such as subgraph isomorphism testing), and they
allow restricted homomorphism dualities, amongst other desirable properties. In
this paper we establish two new characterisations of bounded expansion classes,
one in terms of so-called topological parameters, the other in terms of
controlling dense parts. The latter characterisation is then used to show that
the notion of bounded expansion is compatible with Erd\"os-R\'enyi model of
random graphs with constant average degree. In particular, we prove that for
every fixed , there exists a class with bounded expansion, such that a
random graph of order and edge probability asymptotically almost
surely belongs to the class. We then present several new examples of classes
with bounded expansion that do not exclude some topological minor, and appear
naturally in the context of graph drawing or graph colouring. In particular, we
prove that the following classes have bounded expansion: graphs that can be
drawn in the plane with a bounded number of crossings per edge, graphs with
bounded stack number, graphs with bounded queue number, and graphs with bounded
non-repetitive chromatic number. We also prove that graphs with `linear'
crossing number are contained in a topologically-closed class, while graphs
with bounded crossing number are contained in a minor-closed class
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