TITLE:
Unique Efficient Dominating Sets
AUTHORS:
Isaac Reiter, Ju Zhou
KEYWORDS:
Dominating Set, Minimum Dominating Set, Efficient Dominating Set, Unique Efficient Dominating Set
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.10 No.2,
April
21,
2020
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.