Abstract
During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions.
Highlights
During the last few decades, domination theory has been one of the most active areas of research within graph theory
There are more than 4400 papers already published on domination and related parameters
In the case of total domination, there are over 580 published papers and one book [3]
Summary
During the last few decades, domination theory has been one of the most active areas of research within graph theory. A set S ⊆ V ( G ) is a total dominating set, TDS, of a graph G without isolated vertices if every vertex v ∈ V ( G ) is adjacent to at least one vertex in S.
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