4,896 research outputs found
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Conflict-Free Coloring of Intersection Graphs of Geometric Objects
In FOCS'2002, Even et al. introduced and studied the notion of conflict-free
colorings of geometrically defined hypergraphs. They motivated it by frequency
assignment problems in cellular networks. This notion has been extensively
studied since then.
A conflict-free coloring of a graph is a coloring of its vertices such that
the neighborhood (pointed or closed) of each vertex contains a vertex whose
color differs from the colors of all other vertices in that neighborhood. In
this paper we study conflict-colorings of intersection graphs of geometric
objects. We show that any intersection graph of n pseudo-discs in the plane
admits a conflict-free coloring with O(\log n) colors, with respect to both
closed and pointed neighborhoods. We also show that the latter bound is
asymptotically sharp. Using our methods, we also obtain a strengthening of the
two main results of Even et al. which we believe is of independent interest. In
particular, in view of the original motivation to study such colorings, this
strengthening suggests further applications to frequency assignment in wireless
networks.
Finally, we present bounds on the number of colors needed for conflict-free
colorings of other classes of intersection graphs, including intersection
graphs of axis-parallel rectangles and of \rho-fat objects in the plane.Comment: 18 page
A programme to determine the exact interior of any connected digital picture
Region filling is one of the most important and fundamental operations in
computer graphics and image processing. Many filling algorithms and their
implementations are based on the Euclidean geometry, which are then translated
into computational models moving carelessly from the continuous to the finite
discrete space of the computer. The consequences of this approach is that most
implementations fail when tested for challenging degenerate and nearly
degenerate regions. We present a correct integer-only procedure that works for
all connected digital pictures. It finds all possible interior points, which
are then displayed and stored in a locating matrix. Namely, we present a
filling and locating procedure that can be used in computer graphics and image
processing applications
Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem
In this paper we continue the study started in part I (posted). We consider a
planar, bounded, -connected region , and let \bord\Omega be its
boundary. Let be a cellular decomposition of
\Omega\cup\bord\Omega, where each 2-cell is either a triangle or a
quadrilateral. From these data and a conductance function we construct a
canonical pair where is a special type of a (possibly immersed)
genus singular flat surface, tiled by rectangles and is an energy
preserving mapping from onto . In part I the solution
of a Dirichlet problem defined on was utilized, in this
paper we employ the solution of a mixed Dirichlet-Neumann problem.Comment: 26 pages, 16 figures (color
Coloring Intersection Hypergraphs of Pseudo-Disks
We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks
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