On the parameterized complexity of minimum/maximum degree vertex deletion on several special graphs

In the minimum degree vertex deletion problem, we are given a graph, a distinguished vertex in the graph, and an integer κ, and the question is whether we can delete at most κ vertices from the graph so that the distinguished vertex has the unique minimum degree. The maximum degree vertex deletion problem is defined analogously but here we want the distinguished vertex to have the unique maximum degree. It is known that both problems are NP-hard and fixed-parameter intractable with respect to some natural parameters. In this paper, we study the （parameterized） complexity of these two problems restricted to split graphs, p-degenerate graphs, and planar graphs. Our study provides a comprehensive complexity landscape of the two problems restricted to these special graphs.