94 research outputs found
Density of Range Capturing Hypergraphs
For a finite set of points in the plane, a set in the plane, and a
positive integer , we say that a -element subset of is captured
by if there is a homothetic copy of such that ,
i.e., contains exactly elements from . A -uniform -capturing
hypergraph has a vertex set and a hyperedge set consisting
of all -element subsets of captured by . In case when and
is convex these graphs are planar graphs, known as convex distance function
Delaunay graphs.
In this paper we prove that for any , any , and any convex
compact set , the number of hyperedges in is at most , where is the number of -element
subsets of that can be separated from the rest of with a straight line.
In particular, this bound is independent of and indeed the bound is tight
for all "round" sets and point sets in general position with respect to
.
This refines a general result of Buzaglo, Pinchasi and Rote stating that
every pseudodisc topological hypergraph with vertex set has
hyperedges of size or less.Comment: new version with a tight result and shorter proo
Spectrum of mixed bi-uniform hypergraphs
A mixed hypergraph is a triple , where is
a set of vertices, and are sets of hyperedges. A
vertex-coloring of is proper if -edges are not totally multicolored and
-edges are not monochromatic. The feasible set of is the set of
all integers, , such that has a proper coloring with colors.
Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a
characterization of feasible sets for mixed hypergraphs with all - and
-edges of the same size , .
In this note, we give a short proof of a complete characterization of all
possible feasible sets for mixed hypergraphs with all -edges of size
and all -edges of size , where . Moreover, we show that
for every sequence , , of natural numbers there
exists such a hypergraph with exactly proper colorings using colors,
, and no proper coloring with more than colors. Choosing
this answers a question of Bujt\'as and Tuza, and generalizes
their result with a shorter proof.Comment: 9 pages, 5 figure
Combinatorial and Geometric Properties of Planar Laman Graphs
Laman graphs naturally arise in structural mechanics and rigidity theory.
Specifically, they characterize minimally rigid planar bar-and-joint systems
which are frequently needed in robotics, as well as in molecular chemistry and
polymer physics. We introduce three new combinatorial structures for planar
Laman graphs: angular structures, angle labelings, and edge labelings. The
latter two structures are related to Schnyder realizers for maximally planar
graphs. We prove that planar Laman graphs are exactly the class of graphs that
have an angular structure that is a tree, called angular tree, and that every
angular tree has a corresponding angle labeling and edge labeling.
Using a combination of these powerful combinatorial structures, we show that
every planar Laman graph has an L-contact representation, that is, planar Laman
graphs are contact graphs of axis-aligned L-shapes. Moreover, we show that
planar Laman graphs and their subgraphs are the only graphs that can be
represented this way.
We present efficient algorithms that compute, for every planar Laman graph G,
an angular tree, angle labeling, edge labeling, and finally an L-contact
representation of G. The overall running time is O(n^2), where n is the number
of vertices of G, and the L-contact representation is realized on the n x n
grid.Comment: 17 pages, 11 figures, SODA 201
Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets
A range family is a family of subsets of , like
all halfplanes, or all unit disks. Given a range family , we
consider the -uniform range capturing hypergraphs
whose vertex-sets are finite sets of points
in with any vertices forming a hyperedge whenever for some . Given additionally an integer ,
we seek to find the minimum such that every
admits a polychromatic -coloring of its
vertices, that is, where every hyperedge contains at least one point of each
color. Clearly, and the gold standard is an upper
bound that is linear in .
A -shallow hitting set in is a subset such that for each hyperedge ; i.e.,
every hyperedge is hit at least once but at most times by . We show for
several range families the existence of -shallow hitting sets
in every with being a constant only
depending on . This in particular proves that in such cases, improving previous polynomial bounds in .
Particularly, we prove this for the range families of all axis-aligned strips
in , all bottomless and topless rectangles in , and
for all unit-height axis-aligned rectangles in
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