139,324 research outputs found
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
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
A Central Partition of Molecular Conformational Space.III. Combinatorial Determination of the Volume Spanned by a Molecular System
In the first work of this series [physics/0204035] it was shown that the
conformational space of a molecule could be described to a fair degree of
accuracy by means of a central hyperplane arrangement. The hyperplanes divide
the espace into a hierarchical set of cells that can be encoded by the face
lattice poset of the arrangement. The model however, lacked explicit rotational
symmetry which made impossible to distinguish rotated structures in
conformational space. This problem was solved in a second work
[physics/0404052] by sorting the elementary 3D components of the molecular
system into a set of morphological classes that can be properly oriented in a
standard 3D reference frame. This also made possible to find a solution to the
problem that is being adressed in the present work: for a molecular system
immersed in a heat bath we want to enumerate the subset of cells in
conformational space that are visited by the molecule in its thermal wandering.
If each visited cell is a vertex on a graph with edges to the adjacent cells,
here it is explained how such graph can be built
Light intensity strain analysis
A process is described for the analysis of the strain field of structures subjected to large deformations involving a low modulus substrate having a high modulus, relatively thin coating. The optical properties of transmittance and reflectance are measured for the coated substrate while stressed and unstressed to indicate the strain field for the coated substrate
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