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arXiv · 1705.05911

Four NP-complete problems about generalizations of perfect graphs

Vaidy Sivaraman
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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

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Vaidy Sivaraman. 2017-05-16. Four NP-complete problems about generalizations of perfect graphs. https://arxiv.org/abs/1705.05911

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