53 research outputs found
Manhattan orbifolds
We investigate a class of metrics for 2-manifolds in which, except for a
discrete set of singular points, the metric is locally isometric to an L_1 (or
equivalently L_infinity) metric, and show that with certain additional
conditions such metrics are injective. We use this construction to find the
tight span of squaregraphs and related graphs, and we find an injective metric
that approximates the distances in the hyperbolic plane analogously to the way
the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised
since the previous version, and a new example has been adde
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
Ball and Spindle Convexity with respect to a Convex Body
Let be a convex body. We introduce two notions of
convexity associated to C. A set is -ball convex if it is the
intersection of translates of , or it is either , or . The -ball convex hull of two points is called a -spindle. is
-spindle convex if it contains the -spindle of any pair of its points. We
investigate how some fundamental properties of conventional convex sets can be
adapted to -spindle convex and -ball convex sets. We study separation
properties and Carath\'eodory numbers of these two convexity structures. We
investigate the basic properties of arc-distance, a quantity defined by a
centrally symmetric planar disc , which is the length of an arc of a
translate of , measured in the -norm, that connects two points. Then we
characterize those -dimensional convex bodies for which every -ball
convex set is the -ball convex hull of finitely many points. Finally, we
obtain a stability result concerning covering numbers of some -ball convex
sets, and diametrically maximal sets in -dimensional Minkowski spaces.Comment: 27 pages, 5 figure
On the interplay of combinatorics, geometry, topology and computational complexity
Matematicko-fyzikální fakult
Discrete Geometry
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry
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