A stable cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertice . A cut is stable if all vertices have the (weighted) majority of their neighbors on the other side . We show that the problem is weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable . On the other hand, bounding $tw+\Delta$ is also not enough, as it is NP-Hard for unweighted graphs of bounded degree . We obtain an FPTalgorithm running in time $2^{O(\Delta tw)(n+\log W)^{O(1) and $2^O(tw) is essentially optimal . Unweighted Min StableCut is also probably essentiallyoptimal: an algorithm running in $n^{o(pw) would contradict the ETH. We showed that this is also likely to be a much faster exact algorithm running intime $2 ^O(Tw) or $2\Delta pw\(tw\Delta tw)\Delta tw\Delta Pw\cdotW\(n\Delta W)\Delta)n=O(O(Pw\Delta O(1)\Delta\Delta)\Delta)

**Author(s) :**Michael Lampis

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Keywords : delta - tw - cut - pw - running -