# Parameterized Complexity of Fair Vertex Evaluation Problems

A prototypical graph problem is centered around a graph theoretic property for a set of vertices and a solution is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of certain quality, where we use size as a quality measure. In this work we are changing the measure to the fair measure [Lin&Sahni: Fair edge deletion problems. IEEE Trans. Comput. 89]. The measure is k if the number of solution neighbors does not exceed k for any vertex in the graph. One possible way to study graph problems is by defining the property in a certain logic. For a given objective an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. In the presented paper we show that there is an FPT algorithm for the MSO Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph. Here, the free variable corresponds to the solution sought. One may define an extended variant of MSO Fair Vertex Evaluation for formulas with l free variables; here we measure maximum number of neighbors in each of the l sets. However, such variant is W[1]-hard even on graphs with twin cover one. Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is W[1]-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width.

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