722 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
Liftings and stresses for planar periodic frameworks
We formulate and prove a periodic analog of Maxwell's theorem relating
stressed planar frameworks and their liftings to polyhedral surfaces with
spherical topology. We use our lifting theorem to prove deformation and
rigidity-theoretic properties for planar periodic pseudo-triangulations,
generalizing features known for their finite counterparts. These properties are
then applied to questions originating in mathematical crystallography and
materials science, concerning planar periodic auxetic structures and ultrarigid
periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual
Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201
Inconstancy of finite and infinite sequences
In order to study large variations or fluctuations of finite or infinite
sequences (time series), we bring to light an 1868 paper of Crofton and the
(Cauchy-)Crofton theorem. After surveying occurrences of this result in the
literature, we introduce the inconstancy of a sequence and we show why it seems
more pertinent than other criteria for measuring its variational complexity. We
also compute the inconstancy of classical binary sequences including some
automatic sequences and Sturmian sequences.Comment: Accepted by Theoretical Computer Scienc
Drawing Graphs as Spanners
We study the problem of embedding graphs in the plane as good geometric
spanners. That is, for a graph , the goal is to construct a straight-line
drawing of in the plane such that, for any two vertices and
of , the ratio between the minimum length of any path from to
and the Euclidean distance between and is small. The maximum such
ratio, over all pairs of vertices of , is the spanning ratio of .
First, we show that deciding whether a graph admits a straight-line drawing
with spanning ratio , a proper straight-line drawing with spanning ratio
, and a planar straight-line drawing with spanning ratio are
NP-complete, -complete, and linear-time solvable problems,
respectively, where a drawing is proper if no two vertices overlap and no edge
overlaps a vertex.
Second, we show that moving from spanning ratio to spanning ratio
allows us to draw every graph. Namely, we prove that, for every
, every (planar) graph admits a proper (resp. planar) straight-line
drawing with spanning ratio smaller than .
Third, our drawings with spanning ratio smaller than have large
edge-length ratio, that is, the ratio between the length of the longest edge
and the length of the shortest edge is exponential. We show that this is
sometimes unavoidable. More generally, we identify having bounded toughness as
the criterion that distinguishes graphs that admit straight-line drawings with
constant spanning ratio and polynomial edge-length ratio from graphs that
require exponential edge-length ratio in any straight-line drawing with
constant spanning ratio
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