Abstract
Given a finite simple graph G, a set D ⊆ V(G) is called a dominating set if for all v ∈ V(G) , either v ∈ D or v is adjacent to some vertex in D. A dominating set D is independent if none of the vertices in D are adjacent, and D is perfect if each vertex not in D is adjacent to precisely one vertex in D. If a dominating set is both independent and perfect, then it is called an efficient dominating set. For a graph G, a set D is called a unique efficient dominating set of G if it is the only efficient dominating set of G. In this paper, the authors propose the definition of unique efficient dominating set, explore the properties of graphs with unique efficient dominating sets, and completely characterize several families of graphs which have unique efficient dominating sets.
Highlights
Let G be a simple finite graph and u,v be two vertices of G
The authors propose the definition of unique efficient dominating set, explore the properties of graphs with unique efficient dominating sets, and completely characterize several families of graphs which have unique efficient dominating sets
The following results provide a comprehensive summary of efficient dominating sets of spider graphs, which is a specific type of tree
Summary
Let G be a simple finite graph and u,v be two vertices of G. If a dominating set is both independent and perfect, it is called an efficient dominating set. The following theorem, which was proved in [1], establishes the fact that an efficient dominating set is a minimum dominating set. The following lemma applies to all nontrivial efficient dominating sets. I) It follows directly from the fact that an efficient dominating set is independent and perfect.
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