Complexity of Dominating Induced Matching and Perfect Edge Domination Problems for Neighborhood Star-Free Graphs

Основные понятия

The paper studies the complexity of the dominating induced matching (DIM) problem and the perfect edge domination (PED) problem for neighborhood star-free (NSF) graphs. It proves that the corresponding decision problems are NP-Complete for several subclasses of NSF graphs.

Аннотация

The paper focuses on neighborhood star-free (NSF) graphs, where every vertex of degree at least 2 is contained in a triangle. It explores the complexity of two graph problems on this class of graphs:

Dominating Induced Matching (DIM) problem: Determining whether a graph contains a dominating induced matching.
Perfect Edge Domination (PED) problem: Determining whether a graph contains a perfect edge dominating set.

The key insights and findings are:

The authors prove that deciding if a connected NSF graph contains a DIM is an NP-Complete problem. They describe NP-Completeness proofs for several subclasses of connected NSF graphs.
They show that connected NSF graphs do not have any proper perfect dominating sets, and the only possible PEDs are either the trivial PED or the EEDs.
The paper introduces five variants of the 1in3SAT problem, two of which are known to be NP-Complete, and proves that the other three are also NP-Complete.
The authors provide polynomial-time reductions from these 1in3SAT variants to the existence of DIMs on certain subclasses of connected NSF graphs, establishing the NP-Completeness of the DIM problem for these subclasses.
The paper also discusses the complexity of the PED problem for the same subclasses of connected NSF graphs.
Finally, the authors pose an open question about the potential algorithmic relationship between efficient and perfect edge domination problems.

Статистика

None

Цитаты

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Ключевые выводы из

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

by Vinicius L. ... в arxiv.org 04-09-2024

https://arxiv.org/pdf/2204.11785.pdf

Дополнительные вопросы

Are there any graph classes for which the perfect edge domination problem can be solved in polynomial time while the efficient edge domination problem remains NP-Complete

In the context of graph theory, there are instances where the perfect edge domination problem can be solved in polynomial time while the efficient edge domination problem remains NP-Complete. One such example is when considering certain hereditary graph classes. For instance, in the case of bipartite graphs, the perfect edge domination problem is known to be NP-Complete, while the efficient edge domination problem can be solved in polynomial time. This discrepancy arises due to the specific structural properties and characteristics of bipartite graphs that allow for more efficient solutions to the efficient edge domination problem compared to the perfect edge domination problem.

What other structural properties of neighborhood star-free graphs could be exploited to develop efficient algorithms for the dominating induced matching and perfect edge domination problems

In neighborhood star-free graphs, several structural properties can be exploited to develop efficient algorithms for the dominating induced matching and perfect edge domination problems. One key property is the absence of certain induced cycles, such as C4, which can simplify the identification of dominating induced matchings. By leveraging the unique characteristics of neighborhood star-free graphs, algorithms can be designed to efficiently determine the existence of dominating induced matchings and perfect edge domination sets. Additionally, the presence of triangles in these graphs can be utilized to optimize the identification of dominating sets and matchings, as triangles play a crucial role in the connectivity and domination properties of the graph.

How do the complexity results presented in this paper relate to the broader understanding of the computational complexity of domination problems in graph theory

The complexity results presented in the paper shed light on the computational complexity of domination problems in graph theory, particularly in the context of neighborhood star-free graphs. By establishing the NP-Completeness of the dominating induced matching and perfect edge domination problems for subclasses of connected NSF graphs, the paper contributes to a deeper understanding of the challenges involved in solving these problems. These results highlight the intricate relationships between graph structures, domination properties, and computational complexity, providing valuable insights into the algorithmic complexity of domination problems in diverse graph classes. The findings presented in the paper can serve as a foundation for further research in algorithm design and complexity analysis in graph theory.