56 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
Hierarchical Partial Planarity
In this paper we consider graphs whose edges are associated with a degree of
{\em importance}, which may depend on the type of connections they represent or
on how recently they appeared in the scene, in a streaming setting. The goal is
to construct layouts of these graphs in which the readability of an edge is
proportional to its importance, that is, more important edges have fewer
crossings. We formalize this problem and study the case in which there exist
three different degrees of importance. We give a polynomial-time testing
algorithm when the graph induced by the two most important sets of edges is
biconnected. We also discuss interesting relationships with other
constrained-planarity problems.Comment: Conference version appeared in WG201
The Complexity of Simultaneous Geometric Graph Embedding
Given a collection of planar graphs on the same set of
vertices, the simultaneous geometric embedding (with mapping) problem, or
simply -SGE, is to find a set of points in the plane and a bijection
such that the induced straight-line drawings of
under are all plane.
This problem is polynomial-time equivalent to weak rectilinear realizability
of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x)
proved to be complete for , the existential theory of the
reals. Hence the problem -SGE is polynomial-time equivalent to several other
problems in computational geometry, such as recognizing intersection graphs of
line segments or finding the rectilinear crossing number of a graph.
We give an elementary reduction from the pseudoline stretchability problem to
-SGE, with the property that both numbers and are linear in the
number of pseudolines. This implies not only the -hardness
result, but also a lower bound on the minimum size of a
grid on which any such simultaneous embedding can be drawn. This bound is
tight. Hence there exists such collections of graphs that can be simultaneously
embedded, but every simultaneous drawing requires an exponential number of bits
per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is
only
Simultaneous Embeddability of Two Partitions
We study the simultaneous embeddability of a pair of partitions of the same
underlying set into disjoint blocks. Each element of the set is mapped to a
point in the plane and each block of either of the two partitions is mapped to
a region that contains exactly those points that belong to the elements in the
block and that is bounded by a simple closed curve. We establish three main
classes of simultaneous embeddability (weak, strong, and full embeddability)
that differ by increasingly strict well-formedness conditions on how different
block regions are allowed to intersect. We show that these simultaneous
embeddability classes are closely related to different planarity concepts of
hypergraphs. For each embeddability class we give a full characterization. We
show that (i) every pair of partitions has a weak simultaneous embedding, (ii)
it is NP-complete to decide the existence of a strong simultaneous embedding,
and (iii) the existence of a full simultaneous embedding can be tested in
linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201
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