Question 4: In the DISJOINT CLUSTER VERTEX DELETION problem, if the s…

A graph is called a cluster graph if and only if it is a disjoint union of cliques.

The CLUSTER VERTEX DELETION (CVD) problem is the following: Given an input graph $G$ and a parameter $k$, does there exist a set $S$ of at most $k$ vertices of $G$ such that the subgraph induced on $V(G) \setminus S$ is a cluster graph.

In the DISJOINT CLUSTER VERTEX DELETION (DCVD) problem we are given an input graph $G$, parameter $k$ and a set $S$ of vertices of $G$ of size $k+1$ such that the subgraph induced on $V(G) \setminus S$ is a cluster graph and the subgraph $G[S]$ is also a cluster graph. Our goal is to determine if there exists a subset $T \subset V(G)$ of size at most $k$ which is disjoint from $S$ such that the subgraph induced on $V(G) \setminus T$ is a cluster graph.

Answer the given subquestions about CVD and DCVD.

In the DISJOINT CLUSTER VERTEX DELETION problem, if the subgraph induced on the vertices of S is not a cluster graph, we can immediately return NO.