Parameterized complexity of fair deletion problems

Deletion problems are those where given a graph $G$ and a graph property $\pi$, the goal is to find a subset of edges such that after its removal the graph $G$ will satisfy the property $\pi$. Typically, we want to minimize the number of elements removed. In fair deletion problems we change the objective: we minimize the maximum number of deletions in a neighborhood of a single vertex. We study the parameterized complexity of fair deletion problems with respect to the structural parameters of the tree-width, the path-width, the size of a minimum feedback vertex set, the neighborhood diversity, and the size of minimum vertex cover of graph $G$. We prove the W[1]-hardness of the fair FO vertex-deletion problem with respect to the first three parameters combined. Moreover, we show that there is no algorithm for fair FO vertex-deletion problem running in time $n^{o(k^{1/3})}$, where $n$ is the size of the graph and $k$ is the sum of the first three mentioned parameters, provided that the Exponential Time Hypothesis holds. On the other hand, we provide an FPT algorithm for the fair MSO edge-deletion problem parameterized by the size of minimum vertex cover and an FPT algorithm for the fair MSO vertex-deletion problem parameterized by the neighborhood diversity