1,610 research outputs found
Packing Plane Spanning Trees and Paths in Complete Geometric Graphs
We consider the following question: How many edge-disjoint plane spanning
trees are contained in a complete geometric graph on any set of
points in general position in the plane? We show that this number is in
. Further, we consider variants of this problem by bounding
the diameter and the degree of the trees (in particular considering spanning
paths).Comment: This work was presented at the 26th Canadian Conference on
Computational Geometry (CCCG 2014), Halifax, Nova Scotia, Canada, 2014. The
journal version appeared in Information Processing Letters, 124 (2017),
35--4
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids
The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest
integer k such that for every colouring of the vertices of H with exactly k
colours, there is a hyperedge of H all of whose vertices have different
colours. We denote by nu(H) the number of vertices of H and by tau(H) the size
of the smallest set containing at least two vertices of each hyperedge of H.
For a complete geometric graph G with n > 2 vertices let H = H(G) be the
hypergraph whose vertices are the edges of G and whose hyperedges are the edge
sets of plane spanning trees of G. We prove that if G has at most one interior
vertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) -
tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given
by the ground set and the bases of a matroid, respectively
Packing Short Plane Spanning Trees in Complete Geometric Graphs
Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is very long (when compared to the minimum length needed to obtain a spanning graph). We consider two different approaches: first we show an almost optimal centralized approach to extract two trees. Then we show a constant factor approximation for a distributed model in which each point can compute its adjacencies using only local information. This second approach may create cycles, but maintains planarity
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Packing Plane Spanning Trees into a Point Set
Let be a set of points in the plane in general position. We show that
at least plane spanning trees can be packed into the
complete geometric graph on . This improves the previous best known lower
bound . Towards our proof of this lower bound we
show that the center of a set of points, in the -dimensional space in
general position, is of dimension either or
Three Edge-disjoint Plane Spanning Paths in a Point Set
We study the following problem: Given a set of points in the plane,
how many edge-disjoint plane straight-line spanning paths of can one draw?
A well known result is that when the points are in convex position,
such paths always exist, but when the points of are in
general position the only known construction gives rise to two edge-disjoint
plane straight-line spanning paths. In this paper, we show that for any set
of at least ten points, no three of which are collinear, one can draw at least
three edge-disjoint plane straight-line spanning paths of~. Our proof is
based on a structural theorem on halving lines of point configurations and a
strengthening of the theorem about two spanning paths, which we find
interesting in its own right: if has at least six points, and we prescribe
any two points on the boundary of its convex hull, then the set contains two
edge-disjoint plane spanning paths starting at the prescribed points.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
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