158 research outputs found
Cop and robber game and hyperbolicity
In this note, we prove that all cop-win graphs G in the game in which the
robber and the cop move at different speeds s and s' with s'<s, are
\delta-hyperbolic with \delta=O(s^2). We also show that the dependency between
\delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger
condition. This solves an open question from the paper (J. Chalopin et al., Cop
and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25
(2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and
s'=r+2\delta for any r>0, this establishes a new - game-theoretical -
characterization of Gromov hyperbolicity. We also show that for weakly modular
graphs the dependency between \delta and s is linear for any s'<s. Using these
results, we describe a simple constant-factor approximation of the
hyperbolicity \delta of a graph on n vertices in O(n^2) time when the graph is
given by its distance-matrix
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
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
- …