Clustering to Given Connectivities
Résumé
We define a general variant of the graph clustering problem where
the criterion of density for the clusters is (high) connectivity. In \probCGClong,
we are given an $n$-vertex graph $G$, an integer $k$, and a sequence $\Lambda=\langle \lambda_{1},\ldots,\lambda_{t}\rangle$ of positive integers and we ask whether it is possible to remove at most $k$ edges from $G$ such that the resulting connected
components are {\sl exactly} $t$ and their corresponding edge connectivities are lower-bounded by
the numbers in $\Lambda$.
We prove that this problem, parameterized by $k$, is fixed parameter tractable i.e., can be solved by an $f(k)\cdot n^{O(1)}$-step algorithm, for some function $f$
that depends only on the parameter $k$. Our algorithm
uses the recursive understanding technique that is especially adapted so to
deal with the fact that
we do not impose any restriction
to the connectivity demands in $\Lambda$.
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