  We show that the following problems are NP-complete.
  1. Can the vertex set of a graph be partitioned into two sets such that each
set induces a perfect graph?
  2. Is the difference between the chromatic number and clique number at most
$1$ for every induced subgraph of a graph?
  3. Can the vertex set of every induced subgraph of a graph be partitioned
into two sets such that the first set induces a perfect graph, and the clique
number of the graph induced by the second set is smaller than that of the
original induced subgraph?
  4. Does a graph contain a stable set whose deletion results in a perfect
graph?
  The proofs of the NP-completeness of the four problems follow the same
pattern: Showing that all the four problems are NP-complete when restricted to
triangle-free graphs by using results of Maffray and Preissmann on
$3$-colorability and $4$-colorability of triangle-free graphs
