1,024 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
Core congestion is inherent in hyperbolic networks
We investigate the impact the negative curvature has on the traffic
congestion in large-scale networks. We prove that every Gromov hyperbolic
network admits a core, thus answering in the positive a conjecture by
Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which
is based on the experimental observation by Narayan and Saniee, Physical Review
E, 84 (2011) that real-world networks with small hyperbolicity have a core
congestion. Namely, we prove that for every subset of vertices of a
-hyperbolic graph there exists a vertex of such that the
disk of radius centered at intercepts at least
one half of the total flow between all pairs of vertices of , where the flow
between two vertices is carried by geodesic (or quasi-geodesic)
-paths. A set intercepts the flow between two nodes and if
intersect every shortest path between and . Differently from what
was conjectured by Jonckheere et al., we show that is not (and cannot be)
the center of mass of but is a node close to the median of in the
so-called injective hull of . In case of non-uniform traffic between nodes
of (in this case, the unit flow exists only between certain pairs of nodes
of defined by a commodity graph ), we prove a primal-dual result showing
that for any the size of a -multi-core (i.e., the number
of disks of radius ) intercepting all pairs of is upper bounded by
the maximum number of pairwise -apart pairs of
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