2,161 research outputs found
The Minimum Wiener Connector
The Wiener index of a graph is the sum of all pairwise shortest-path
distances between its vertices. In this paper we study the novel problem of
finding a minimum Wiener connector: given a connected graph and a set
of query vertices, find a subgraph of that connects all
query vertices and has minimum Wiener index.
We show that The Minimum Wiener Connector admits a polynomial-time (albeit
impractical) exact algorithm for the special case where the number of query
vertices is bounded. We show that in general the problem is NP-hard, and has no
PTAS unless . Our main contribution is a
constant-factor approximation algorithm running in time
.
A thorough experimentation on a large variety of real-world graphs confirms
that our method returns smaller and denser solutions than other methods, and
does so by adding to the query set a small number of important vertices
(i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International
Conference on Management of Dat
Inverse Problems Related to the Wiener and Steiner-Wiener Indices
In a graph, the generalized distance between multiple vertices is the minimum number of edges in a connected subgraph that contains these vertices. When we consider such distances between all subsets of vertices and take the sum, it is called the Steiner -Wiener index and has important applications in Chemical Graph Theory. In this thesis we consider the inverse problems related to the Steiner Wiener index, i.e. for what positive integers is there a graph with Steiner Wiener index of that value
Steiner Distance in Product Networks
For a connected graph of order at least and , the
\emph{Steiner distance} among the vertices of is the minimum size
among all connected subgraphs whose vertex sets contain . Let and be
two integers with . Then the \emph{Steiner -eccentricity
} of a vertex of is defined by . Furthermore, the
\emph{Steiner -diameter} of is . In this paper, we investigate the Steiner distance and Steiner
-diameter of Cartesian and lexicographical product graphs. Also, we study
the Steiner -diameter of some networks.Comment: 29 pages, 4 figure
A Central Limit Theorem for the Poisson-Voronoi Approximation
For a compact convex set and a Poisson point process , the union of
all Voronoi cells with a nucleus in is the Poisson-Voronoi approximation of
. Lower and upper bounds for the variance and a central limit theorem for
the volume of the Poisson-Voronoi approximation are shown. The proofs make use
of so called Wiener-It\^o chaos expansions and the central limit theorem is
based on a more abstract central limit theorem for Poisson functionals, which
is also derived.Comment: 22 pages, modified reference
Wiener Index and Remoteness in Triangulations and Quadrangulations
Let be a a connected graph. The Wiener index of a connected graph is the
sum of the distances between all unordered pairs of vertices. We provide
asymptotic formulae for the maximum Wiener index of simple triangulations and
quadrangulations with given connectivity, as the order increases, and make
conjectures for the extremal triangulations and quadrangulations based on
computational evidence. If denotes the arithmetic mean
of the distances from to all other vertices of , then the remoteness of
is defined as the largest value of over all vertices
of . We give sharp upper bounds on the remoteness of simple
triangulations and quadrangulations of given order and connectivity
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