Total Domination in Graphs with Diameter 2

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Abstract

The total domination number inline image of a graph G is the minimum cardinality of a set S of vertices, so that every vertex of G is adjacent to a vertex in S. In this article, we determine an optimal upper bound on the total domination number of a graph with diameter 2. We show that for every graph G on n vertices with diameter 2, inline image. This bound is optimal in the sense that given any inline image, there exist graphs G with diameter 2 of all sufficiently large even orders n such that inline image.

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