Graphs with few total dominating sets

doi:10.1016/j.disc.2018.01.006

Discrete Mathematics

Volume 341, Issue 4, April 2018, Pages 997-1009

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Abstract

We give a lower bound for the number of total dominating sets of a graph together with a characterization of the extremal graphs, for trees as well as arbitrary connected graphs of given order. Moreover, we obtain a sharp lower bound involving both the order and the total domination number, and characterize the extremal graphs as well.

Keywords

Total dominating set

Total domination number

Subdivided star

Lower bound

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¹: Research fellow of the Claude Leon Foundation at the Department of Pure and Applied Mathematics, University of Johannesburg, South Africa.