In this paper we study fair variants of MSO$_1$ definable problems parameterized by cluster vertex deletion number, i.e., the smallest number of vertices required to be removed from the graph such that what remains is a collection of cliques. While typical graph problems seek the smallest set of vertices satisfying some property, their fair variants seek such a set that does not contain too many vertices in any neighborhood of any vertex. Formally, the task is to find a set $X\subseteq V(G)$ satisfying some MSO$_1$ definable property, whose fair cost is at most $k$, i.e., such that for all $v\in V(G)$ it holds that $|X\cap N(v)|\le k$. Recently, Knop, Masařík, and Toufar [MFCS 2019] showed that all fair MSO$_1$ definable problems can be solved in FPT time parameterized by the twin cover of a graph. They asked whether such a statement can be achieved for a more general parameterization by cluster vertex deletion number. In this paper, we prove that in full generality this is not possible by demonstrating W[1]-hardness. On the other hand, we give a sufficient condition under which a fair MSO$_1$ definable problem admits an FPT algorithm parameterized by the cluster vertex deletion number. Our algorithm is general enough to capture the fair variant of many natural graph problems such as the Fair Feedback Vertex Set problem, the Fair Vertex Cover problem, the Fair Dominating Set problem, the Fair Odd Cycle Transversal problem, as well as connected variants thereof. Moreover, we solve the Fair $[σ,ρ]$-Domination problem for $σ$ finite, or when both $σ$ and $ρ$ are cofinite. That is, given finite or cofinite $ρ,σ\subseteq \mathbb{N}$, the task is to find set of vertices $X\subseteq V(G)$ of fair cost at most $k$ such that for all $v\in X$, $|N(v)\cap X| \inσ$ and for all $v\in V(G)\setminus X$, $|N(v)\cap X|\inρ$.

翻译：本文研究由MSO$_1$可定义问题在团簇顶点删除数参数化下的公平变体，即从图中移除的最小顶点数使得剩余部分为团簇集合。典型图问题寻求满足某性质的最小顶点集，而其公平变体则要求该集合在任何顶点的邻域中不包含过多顶点。形式化地，任务为寻找满足某MSO$_1$可定义性质的顶点集$X\subseteq V(G)$，其公平代价不超过$k$，即对所有$v\in V(G)$满足$|X\cap N(v)|\le k$。近期，Knop、Masařík和Toufar [MFCS 2019]证明了所有公平MSO$_1$可定义问题可在图的双覆盖参数化下以FPT时间求解。他们提出是否能在团簇顶点删除数的更一般参数化下实现此类结果。本文证明在完全一般性下这是不可能的，并展示了W[1]-困难性。另一方面，我们给出充分条件使得公平MSO$_1$可定义问题在团簇顶点删除数参数化下存在FPT算法。该算法足够通用，可涵盖自然图问题的公平变体，如公平反馈顶点集问题、公平顶点覆盖问题、公平支配集问题、公平奇圈横贯问题及其连通变体。此外，我们针对有限$\sigma$或$\sigma$与$\rho$共有限的情况，解决了公平$[\sigma,\rho]$-支配问题。即给定有限或共有限$\rho,\sigma\subseteq \mathbb{N}$，任务为寻找公平代价不超过$k$的顶点集$X\subseteq V(G)$，使得对所有$v\in X$有$|N(v)\cap X| \in\sigma$，且对所有$v\in V(G)\setminus X$有$|N(v)\cap X|\in\rho$。