核心概念
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.
摘要
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.