  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 is
contained in a triangle except pendant vertices and isolated vertices. This
class has been considered before for several contexts. In the present paper, we
study the complexity of the dominating induced matching (DIM) problem and the
perfect edge domination (PED) problem for NSF graphs. We prove the
corresponding decision problems are NP-Complete for several of its subclasses.
As an added value of this study, we have shown three connected variants of
planar positive 1in3SAT are also NP-Complete. Since these variants are more
basic in complexity theory context than many graph problems, these results can
be useful to prove that other problems are NP-Complete.
