7 research outputs found
Low Ply Drawings of Trees
We consider the recently introduced model of \emph{low ply graph drawing}, in
which the ply-disks of the vertices do not have many common overlaps, which
results in a good distribution of the vertices in the plane. The
\emph{ply-disk} of a vertex in a straight-line drawing is the disk centered at
it whose radius is half the length of its longest incident edge. The largest
number of ply-disks having a common overlap is called the \emph{ply-number} of
the drawing.
We focus on trees. We first consider drawings of trees with constant
ply-number, proving that they may require exponential area, even for stars, and
that they may not even exist for bounded-degree trees. Then, we turn our
attention to drawings with logarithmic ply-number and show that trees with
maximum degree always admit such drawings in polynomial area.Comment: This is a complete access version of a paper that will appear in the
proceedings of GD201
Contact graphs of line segments are NP-complete
AbstractContact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of line segments in the plane are considered — it is proved that recognizing line-segment contact graphs, with contact degrees of 3 or more, is an NP-complete problem, even for planar graphs. This result contributes to the related research on recognition complexity of curve contact graphs (Hliněný J. Combin. Theory Ser. B 74 (1998) 87)
Coloring non-crossing strings
For a family of geometric objects in the plane
, define as the least
integer such that the elements of can be colored with
colors, in such a way that any two intersecting objects have distinct
colors. When is a set of pseudo-disks that may only intersect on
their boundaries, and such that any point of the plane is contained in at most
pseudo-disks, it can be proven that
since the problem is equivalent to cyclic coloring of plane graphs. In this
paper, we study the same problem when pseudo-disks are replaced by a family
of pseudo-segments (a.k.a. strings) that do not cross. In other
words, any two strings of are only allowed to "touch" each other.
Such a family is said to be -touching if no point of the plane is contained
in more than elements of . We give bounds on
as a function of , and in particular we show that
-touching segments can be colored with colors. This partially answers
a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems
of strings.Comment: 19 pages. A preliminary version of this work appeared in the
proceedings of EuroComb'09 under the title "Coloring a set of touching
strings
Abstracts of theses in mathematics
summary:Krejčíř, Pavel: The theory and applications of spatial statistics and stochastic geometry.
Vítek, Tomáš: Detection of changes in econometric models.
Hliněný, Petr: Contact representations of graphs.
Kliková, Alice: Finite volume -- finite element solution of compressible flow.
Hrach, Karel: Bayesian analysis of models with non-negative residuals.
Svatoš, Jan: M-estimators in the linear model for nonregular densities.
Ševčík, Petr: Extremal martingale measures in finance.
Hlávka, Zdeněk: Robust sequential estimation.
Holub, Štěpán: Equations in free monoids.
Klaschka Jan: Mathematical methods of state change assessment in medical research.
Unzeitigová, Vladimíra: Mathematical models of health insurance for commercial insurance companies -- embedded value of accident insurance.
Friesl, Michal: Bayesian estimation in exponent competing risks and related models with applications to insurance.
Fiala, Jiří: Locally injective homomorphisms.
Kaplický, Petr: Qualitative properties of solutions of systems of mechanics.
Ghoneim, Sobha: Selfdistributive rings and near-rings
Contact Graphs of Curves
Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced, their properties and inclusions between them are studied, and the maximal clique in relation with the chromatic number of contact graphs is considered. Also relations between planar and contact graphs are mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NP--complete (NP--hard) even for planar graphs. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Probably the first type studied were interval graphs (intersection graphs of intervals on a line), owing to their applications in biology, see [14],[1]. We may also mention other kinds of intersection graphs s..
Contact Graphs of Curves (Extended Abstract)
. Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced and their properties and inclusions between them are studied. Also the relation between planar and contact graphs is mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NP--complete (NP--hard) even for planar graphs. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Formally the intersection graph of a set family M is defined as a graph G with the vertex set V (G) = M and the edge set E(G) = \Phi fA; Bg ` M j A 6= B; A " B 6= ; \Psi . Probably the first type studied were interval graphs (intersection graphs of intervals on a line), owing to their applic..