# Four NP-complete problems about generalizations of perfect graphs

## TL;DR

This paper proves four problems related to perfect graphs are NP-complete, expanding understanding of computational complexity in graph theory and perfect graph generalizations.

## Contribution

It establishes NP-completeness for four new problems involving partitions and properties of perfect graphs, using reductions from 3- and 4-colorability of triangle-free graphs.

## Key findings

- Four NP-complete problems about perfect graph generalizations.
- NP-completeness holds even for triangle-free graphs.
- Uses reductions from known coloring problems.

## Abstract

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