Reducing the vertex cover number via edge contractions

Given a graph G on n vertices and two integers k d, the Contraction(vc) problem asks whether one can contract at most edges to reduce vertex cover number of by least d. Recently, Lima et al. [JCSS 2021] proved that admits an XP algorithm running in time f(d)⋅nO(d). They asked this is FPT under parameterization. In article, we prove that: (i) W[1]-hard parameterized k+d. Moreover, unless ETH fails, does not admit f(k+d)⋅no(k+d) for any function f. This answers negatively open question stated 2021]. (ii) NP-hard even when k=d. (iii) be solved 2O(d)⋅nk−d+O(1). improves 2021], shows k=d, d (or k).