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

¹ Departamento de Matemática, Universidade Federal Rural do Rio de Janeiro, Seropédica, Brazil
² CONICET, Instituto de Cálculo and Departamento de Computación, Universidad de Buenos Aires, Buenos Aires, Argentina
³ Programa de Engenharia de Sistemas e Computação, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
⁴ Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
⁵ Instituto de Ciencias, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina
⁶ Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil

^* Corresponding author: oscarlin@dc.uba.ar

Received: 8 April 2024
Accepted: 16 October 2024

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 show 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.