5,457 research outputs found
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
The Orchard crossing number of an abstract graph
We introduce the Orchard crossing number, which is defined in a similar way
to the well-known rectilinear crossing number. We compute the Orchard crossing
number for some simple families of graphs. We also prove some properties of
this crossing number.
Moreover, we define a variant of this crossing number which is tightly
connected to the rectilinear crossing number, and compute it for some simple
families of graphs.Comment: 17 pages, 10 figures. Totally revised, new material added. Submitte
Manhattan orbifolds
We investigate a class of metrics for 2-manifolds in which, except for a
discrete set of singular points, the metric is locally isometric to an L_1 (or
equivalently L_infinity) metric, and show that with certain additional
conditions such metrics are injective. We use this construction to find the
tight span of squaregraphs and related graphs, and we find an injective metric
that approximates the distances in the hyperbolic plane analogously to the way
the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised
since the previous version, and a new example has been adde
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
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
Mobile vs. point guards
We study the problem of guarding orthogonal art galleries with horizontal
mobile guards (alternatively, vertical) and point guards, using "rectangular
vision". We prove a sharp bound on the minimum number of point guards required
to cover the gallery in terms of the minimum number of vertical mobile guards
and the minimum number of horizontal mobile guards required to cover the
gallery. Furthermore, we show that the latter two numbers can be calculated in
linear time.Comment: This version covers a previously missing case in both Phase 2 &
Approximation Algorithms for Connected Maximum Cut and Related Problems
An instance of the Connected Maximum Cut problem consists of an undirected
graph G = (V, E) and the goal is to find a subset of vertices S V
that maximizes the number of edges in the cut \delta(S) such that the induced
graph G[S] is connected. We present the first non-trivial \Omega(1/log n)
approximation algorithm for the connected maximum cut problem in general graphs
using novel techniques. We then extend our algorithm to an edge weighted case
and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark
contrast to the classical max-cut problem, we show that the connected maximum
cut problem remains NP-hard even on unweighted, planar graphs. On the positive
side, we obtain a polynomial time approximation scheme for the connected
maximum cut problem on planar graphs and more generally on graphs with bounded
genus.Comment: 17 pages, Conference version to appear in ESA 201
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