We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph while deleting exactly~$k$ vertices, and {sc Path-contraction Preserving Strong Connectivity}, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are $sf W[1]$-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable and we provide a $2^{O(klog k)} n^{O(1)}$-algorithm that solves {sc Weighted Biconnectivity Deletion}. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivity-preservation constraints in the parameterized setting.
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