Fair Vertex Problems Parameterized by Cluster Vertex Deletion

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, Masa\v{r}\'ik, 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 $[\sigma,\rho]$-Domination problem for $\sigma$ finite, or when both $\sigma$ and $\rho$ are cofinite. That is, given finite or cofinite $\rho,\sigma\subseteq \mathbb{N}$, the task is to find set of vertices $X\subseteq V(G)$ of fair cost at most $k$ such that for all $v\in X$, $|N(v)\cap X| \in\sigma$ and for all $v\in V(G)\setminus X$, $|N(v)\cap X|\in\rho$.