A note on acyclic domination number in graphs of diameter two (Q2492203)

A dominating set \(S\) in a graph \(G\) (every vertex outside \(S\) has a neighbor in \(S\)) is an acyclic dominating set if the subgraph of \(G\) induced by \(S\) is a forest. The acyclic dominating number \(\gamma_a(G)\) is the size of a smallest acyclic dominating set of \(G\). The minmum degree of \(G\) is denoted by \(\delta(G)\). The authors prove that, for any positive integer \(k\), there exists a graph \(G\) with diameter two such that \(\gamma_a(G) - \delta(G)\geq k\). This is a negative answer to a question posed by \textit{S. M. Hedetniemi} et al. [Discrete Math. 222, 151--165 (2000; Zbl 0961.05052)].