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(0(3 root k)), 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.