Abstract
Given an undirected graph G and an integer d ≥ 0 , the NP-hard Bounded-Degree Vertex Deletion problem asks to delete as few vertices as possible from G such that the resulting graph has maximum vertex degree d . Our main result is to prove that Bounded-Degree Vertex Deletion is W[1]-hard with respect to the parameter treewidth. As a side result, we obtain that the NP-hard Vector Dominating Set problem is W[1]-hard with respect to the parameter treewidth. On the positive side, we show that Bounded-Degree Vertex Deletion becomes fixed-parameter tractable when parameterized by the combined parameter treewidth and number of vertices to delete, and when parametrized by the feedback edge set number.
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