148 research outputs found
Slimness of graphs
Slimness of a graph measures the local deviation of its metric from a tree
metric. In a graph , a geodesic triangle with
is the union of three shortest
paths connecting these vertices. A geodesic triangle is
called -slim if for any vertex on any side the
distance from to is at most , i.e. each path
is contained in the union of the -neighborhoods of two others. A graph
is called -slim, if all geodesic triangles in are
-slim. The smallest value for which is -slim is
called the slimness of . In this paper, using the layering partition
technique, we obtain sharp bounds on slimness of such families of graphs as (1)
graphs with cluster-diameter of a layering partition of , (2)
graphs with tree-length , (3) graphs with tree-breadth , (4)
-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show
that the slimness of every 4-chordal graph is at most 2 and characterize those
4-chordal graphs for which the slimness of every of its induced subgraph is at
most 1
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
A Characterization of Uniquely Representable Graphs
The betweenness structure of a finite metric space is a pair
where is the so-called betweenness
relation of that consists of point triplets such that . The underlying graph of a betweenness structure
is the simple graph where
the edges are pairs of distinct points with no third point between them. A
connected graph is uniquely representable if there exists a unique metric
betweenness structure with underlying graph . It was implied by previous
works that trees are uniquely representable. In this paper, we give a
characterization of uniquely representable graphs by showing that they are
exactly the block graphs. Further, we prove that two related classes of graphs
coincide with the class of block graphs and the class of distance-hereditary
graphs, respectively. We show that our results hold not only for metric but
also for almost-metric betweenness structures.Comment: 16 pages (without references); 3 figures; major changes: simplified
proofs, improved notations and namings, short overview of metric graph theor
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
Markov convexity and nonembeddability of the Heisenberg group
We compute the Markov convexity invariant of the continuous infinite
dimensional Heisenberg group to show that it is Markov
4-convex and cannot be Markov -convex for any . As Markov convexity
is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a
different proof of the classical theorem of Pansu and Semmes that the
Heisenberg group does not admit a biLipschitz embedding into any Euclidean
space.
The Markov convexity lower bound will follow from exhibiting an explicit
embedding of Laakso graphs into that has distortion
at most . We use this to show that if is a Markov
-convex metric space, then balls of the discrete Heisenberg group
of radius embed into with distortion at least
some constant multiple of
Finally, we show that Markov 4-convexity does not give the optimal distortion
for embeddings of binary trees into by showing that
the distortion is on the order of .Comment: version to appear in Ann. Inst. Fourie
Clusters of Cycles
A {\it cluster of cycles} (or {\it -polycycle}) is a simple planar
2--co nnected finite or countable graph of girth and maximal
vertex-degree , which admits {\it -polycyclic realization} on the
plane, denote it by , i.e. such that: (i) all interior vertices are of
degree , (ii) all interior faces (denote their number by ) are
combinatorial -gons and (implied by (i), (ii)) (iii) all vertices, edges and
interior faces form a cell-complex.
An example of -polycycle is the skeleton of , i.e. of the
-valent partition of the sphere , Euclidean plane or hyperbolic
plane by regular -gons. Call {\it spheric} pairs
; for those five pairs is
without the exterior face; otherwise .
We give here a compact survey of results on -polycycles.Comment: 21. to in appear in Journal of Geometry and Physic
Metric characterizations of superreflexivity in terms of word hyperbolic groups and finite graphs
We show that superreflexivity can be characterized in terms of bilipschitz
embeddability of word hyperbolic groups. We compare characterizations of
superreflexivity in terms of diamond graphs and binary trees. We show that
there exist sequences of series-parallel graphs of increasing topological
complexity which admit uniformly bilipschitz embeddings into a Hilbert space,
and thus do not characterize superreflexivity
- …