10,831 research outputs found
Structurally Parameterized d-Scattered Set
In -Scattered Set we are given an (edge-weighted) graph and are asked to
select at least vertices, so that the distance between any pair is at least
, thus generalizing Independent Set. We provide upper and lower bounds on
the complexity of this problem with respect to various standard graph
parameters. In particular, we show the following:
- For any , an -time algorithm, where
is the treewidth of the input graph.
- A tight SETH-based lower bound matching this algorithm's performance. These
generalize known results for Independent Set.
- -Scattered Set is W[1]-hard parameterized by vertex cover (for
edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if
is an additional parameter.
- A single-exponential algorithm parameterized by vertex cover for unweighted
graphs, complementing the above-mentioned hardness.
- A -time algorithm parameterized by tree-depth
(), as well as a matching ETH-based lower bound, both for
unweighted graphs.
We complement these mostly negative results by providing an FPT approximation
scheme parameterized by treewidth. In particular, we give an algorithm which,
for any error parameter , runs in time
and returns a
-scattered set of size , if a -scattered set of the same
size exists
Cluster synchronization in networks of coupled non-identical dynamical systems
In this paper, we study cluster synchronization in networks of coupled
non-identical dynamical systems. The vertices in the same cluster have the same
dynamics of uncoupled node system but the uncoupled node systems in different
clusters are different. We present conditions guaranteeing cluster
synchronization and investigate the relation between cluster synchronization
and the unweighted graph topology. We indicate that two condition play key
roles for cluster synchronization: the common inter-cluster coupling condition
and the intra-cluster communication. From the latter one, we interpret the two
well-known cluster synchronization schemes: self-organization and driving, by
whether the edges of communication paths lie at inter or intra-cluster. By this
way, we classify clusters according to whether the set of edges inter- or
intra-cluster edges are removable if wanting to keep the communication between
pairs of vertices in the same cluster. Also, we propose adaptive feedback
algorithms on the weights of the underlying graph, which can synchronize any
bi-directed networks satisfying the two conditions above. We also give several
numerical examples to illustrate the theoretical results
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