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

Vinicius L. do Forte, Min Chih Lin, Abilio Lucena, Nelson Maculan, Veronica A. Moyano, Jayme L. Szwarcfiter

TL;DR

This paper defines neighborhood star-free ($NSF$) graphs and investigates two edge-domination problems: dominating induced matching ($DIM$) and perfect edge domination ($PED$). It proves NP-Completeness of deciding the existence of a DIM in connected NSF graphs via a sequence of reductions from connected planar positive $1in3SAT$ variants, featuring a Main Transformation $S(G(F))$ and gadget chains to enforce consistent colorings. It extends the hardness to PED (and related PED with at most $m-1$ edges) within the same graph class under multiple forbidden subgraph conditions, while preserving planarity and bounded degree. The work highlights the role of basic SAT variants as hardness sources for structural graph problems and raises open questions about the relationship between efficient and perfect edge domination in non-hereditary graph classes.

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.

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

TL;DR

This paper defines neighborhood star-free () graphs and investigates two edge-domination problems: dominating induced matching () and perfect edge domination (). It proves NP-Completeness of deciding the existence of a DIM in connected NSF graphs via a sequence of reductions from connected planar positive variants, featuring a Main Transformation and gadget chains to enforce consistent colorings. It extends the hardness to PED (and related PED with at most edges) within the same graph class under multiple forbidden subgraph conditions, while preserving planarity and bounded degree. The work highlights the role of basic SAT variants as hardness sources for structural graph problems and raises open questions about the relationship between efficient and perfect edge domination in non-hereditary graph classes.

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.
Paper Structure (3 sections, 9 theorems, 6 figures, 2 tables)

This paper contains 3 sections, 9 theorems, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. NP-Completeness results
  3. Conclusions

Key Result

Theorem 1

CARDOSO2011521 Given a graph $G$ and a bi-coloring $(B,W)$ of $G$. Then $(B,W)$ is valid if and only if it verifies the following conditions.

Figures (6)

  • Figure 1: Some graphs of interests
  • Figure 2: Some wheels
  • Figure 3: Reduction employed in the NP completeness proof
  • Figure 4: An induced $C_4$ cannot exist in the associated graph $G(F')$
  • Figure 5: Replacing a clause by a triangle
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Corollary 1
  • ...and 3 more