213 research outputs found

### Disjoint compatibility graph of non-crossing matchings of points in convex position

Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
$X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do
not have common edges, and no edge of $M$ crosses an edge of $M'$. Denote by
$\mathrm{DCM}_k$ the graph whose vertices correspond to such matchings, and two
vertices are adjacent if and only if the corresponding matchings are disjoint
compatible. We show that for each $k \geq 9$, the connected components of
$\mathrm{DCM}_k$ form exactly three isomorphism classes -- namely, there is a
certain number of isomorphic small components, a certain number of isomorphic
medium components, and one big component. The number and the structure of small
and medium components is determined precisely.Comment: 46 pages, 30 figure

### On the Number of Pseudo-Triangulations of Certain Point Sets

We pose a monotonicity conjecture on the number of pseudo-triangulations of
any planar point set, and check it on two prominent families of point sets,
namely the so-called double circle and double chain. The latter has
asymptotically $12^n n^{\Theta(1)}$ pointed pseudo-triangulations, which lies
significantly above the maximum number of triangulations in a planar point set
known so far.Comment: 31 pages, 11 figures, 4 tables. Not much technical changes with
respect to v1, except some proofs and statements are slightly more precise
and some expositions more clear. This version has been accepted in J. Combin.
Th. A. The increase in number of pages from v1 is mostly due to formatting
the paper with "elsart.cls" for Elsevie

### Triangulations without pointed spanning trees

Problem 50 in the Open Problems Project asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph. We provide a counterexample. As a consequence we show that there exist
triangulations which require a linear number of edge flips to become Hamiltonian.Acciones Integradas 2003-2004Austrian Fonds zur Förderung der Wissenschaftlichen Forschun

### Different Types of Isomorphisms of Drawings of Complete Multipartite Graphs

Simple drawings are drawings of graphs in which any two edges intersect at
most once (either at a common endpoint or a proper crossing), and no edge
intersects itself. We analyze several characteristics of simple drawings of
complete multipartite graphs: which pairs of edges cross, in which order they
cross, and the cyclic order around vertices and crossings, respectively. We
consider all possible combinations of how two drawings can share some
characteristics and determine which other characteristics they imply and which
they do not imply. Our main results are that for simple drawings of complete
multipartite graphs, the orders in which edges cross determine all other
considered characteristics. Further, if all partition classes have at least
three vertices, then the pairs of edges that cross determine the rotation
system and the rotation around the crossings determine the extended rotation
system. We also show that most other implications -- including the ones that
hold for complete graphs -- do not hold for complete multipartite graphs. Using
this analysis, we establish which types of isomorphisms are meaningful for
simple drawings of complete multipartite graphs.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023

### Linear transformation distance for bichromatic matchings

Let $P=B\cup R$ be a set of $2n$ points in general position, where $B$ is a
set of $n$ blue points and $R$ a set of $n$ red points. A \emph{$BR$-matching}
is a plane geometric perfect matching on $P$ such that each edge has one red
endpoint and one blue endpoint. Two $BR$-matchings are compatible if their
union is also plane.
The \emph{transformation graph of $BR$-matchings} contains one node for each
$BR$-matching and an edge joining two such nodes if and only if the
corresponding two $BR$-matchings are compatible. In SoCG 2013 it has been shown
by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is
always connected, but its diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of the transformation graph
and prove an upper bound of $2n$ for its diameter, which is asymptotically
tight

### Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1)
$\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex
vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices
of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
$\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of $\mathcal{P}$

### Empty Monochromatic Simplices

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq
3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common
$(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic
$d$-simplices determined by $S$, that is, simplices which have only points from
one color class of $S$ as vertices and no points of $S$ in their interior. For
$3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and
strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various
results on triangulations of point sets in $\mathbb{R}^d$. In particular, for
any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$
sufficiently large), in general position in $\mathbb{R}^d$, admits a
triangulation with at least $dn+\Omega(\log n)$ simplices

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