Core Concepts
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.
Abstract
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.