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

TL;DR

This paper investigates the computational complexity of dominating induced matching and perfect edge domination problems within neighborhood star-free graphs, establishing NP-Completeness for several subclasses and connecting these results to variants of planar positive 1in3SAT.

Contribution

It demonstrates NP-Completeness of key domination problems in NSF graphs and introduces NP-Completeness results for three variants of planar positive 1in3SAT.

Findings

DIM and PED problems are NP-Complete for NSF subclasses

Three variants of planar positive 1in3SAT are NP-Complete

Results aid in proving NP-Completeness of other problems

Abstract

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.