On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs GG satisfying the following triangle condition: for every maximal independent set II in GG and every edge uvuv in G−IG−I, there is a vertex w∈Iw∈I such that {u,v,w}{u,v,w} is a triangle in GG. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NPNP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NPP=NP.