734 research outputs found
Polyhedra, Complexes, Nets and Symmetry
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are
finite or infinite 3-periodic structures with interesting geometric,
combinatorial, and algebraic properties. They can be viewed as finite or
infinite 3-periodic graphs (nets) equipped with additional structure imposed by
the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is
"regular" if its geometric symmetry group is transitive on the flags (incident
vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra
and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all
infinite, which are not polyhedra. Their edge graphs are nets well-known to
crystallographers, and we identify them explicitly. There also are 6 infinite
families of "chiral" apeirohedra, which have two orbits on the flags such that
adjacent flags lie in different orbits.Comment: Acta Crystallographica Section A (to appear
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Enumeration of labelled 4-regular planar graphs
We present the first combinatorial scheme for counting labelled 4-regular
planar graphs through a complete recursive decomposition. More precisely, we
show that the exponential generating function of labelled 4-regular planar
graphs can be computed effectively as the solution of a system of equations,
from which the coefficients can be extracted. As a byproduct, we also enumerate
labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps
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