The complexity of some problems related to Graph 3-colorability Original Research Article

It is well-known that the Graph 3-colorability problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in lt]o li](1) a tree or li](2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given graph has a stable set whose deletion results in li](3) the complement of a bipartite graph, li](4) a split graph or li](5) a threshold graph.