2,700 research outputs found
Laplacian Distribution and Domination
Let denote the number of Laplacian eigenvalues of a graph in an
interval , and let denote its domination number. We extend the
recent result , and show that isolate-free graphs also
satisfy . In pursuit of better understanding Laplacian
eigenvalue distribution, we find applications for these inequalities. We relate
these spectral parameters with the approximability of , showing that
. However, for -cyclic graphs, . For trees ,
Dominating Set Games
In this paper we study cooperative cost games arising from domination problems on graphs.We introduce three games to model the cost allocation problem and we derive a necessary and su cient condition for the balancedness of all three games.Furthermore we study concavity of these games.game theory;cost allocation;cooperative games
Distributed Connectivity Decomposition
We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
(I) A decomposition of each undirected graph with vertex-connectivity
into (fractionally) vertex-disjoint weighted dominating trees with total weight
, in rounds.
(II) A decomposition of each undirected graph with edge-connectivity
into (fractionally) edge-disjoint weighted spanning trees with total
weight , in
rounds.
We also show round complexity lower bounds of
and
for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from
to near-optimal .
As corollaries, we also get distributed oblivious routing broadcast with
-competitive edge-congestion and -competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal -approximation of vertex connectivity: centralized
and distributed . The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation
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