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A pendant vertex is one of degree one and an isolated vertex has degree zero.A neighborhood star-free (NSF for short) graph is one in which every vertex iscontained in a triangle except pendant vertices and isolated vertices. Thisclass has been considered before for several contexts. In the present paper, westudy the complexity of the dominating induced matching (DIM) problem and theperfect edge domination (PED) problem for NSF graphs. We prove thecorresponding decision problems are NP-Complete for several of its subclasses.As an added value of this study, we have shown three connected variants ofplanar positive 1in3SAT are also NP-Complete. Since these variants are morebasic in complexity theory context than many graph problems, these results canbe useful to prove that other problems are NP-Complete.

Graphs whose vertices of degree at least 2 lie in a triangle