OJDM  Vol.10 No.2 , April 2020
Unique Efficient Dominating Sets
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.
Cite this paper: Reiter, I. and Zhou, J. (2020) Unique Efficient Dominating Sets. Open Journal of Discrete Mathematics, 10, 56-68. doi: 10.4236/ojdm.2020.102006.

[1]   Mostaghim, Z. and Kjalkhali, A.S. (2012) Efficient Dominating Sets in Ladder Graphs. International Journal of Engineering Research and Development, 2, 42-43.

[2]   Hedetniemi, J. (2017) On Graphs Having a Unique Minimum Independent Dominating Set. Australian Journal of Combinatorics, 68, 357-370.

[3]   Wolfram MathWorld (2019).