The Sierpiński domination number (Q6617234)

An interesting generalization of Sierpiński graphs has recently been proposed by \textit{J. Kovič} et al. [ibid. 23, No. 1, Paper No. 1, 25 p. (2023; Zbl 1504.05244)]. They introduced the Sierpiński product of graphs as follows. Let \(G\) and \(H\) be graphs and let \(f:V(G)\rightarrow V(H)\) be an arbitrary function. The Sierpiński product of graphs \(G\) and \(H\) with respect to \(f\), denoted by \(G\otimes_f H\), is defined as the graph on the vertex set \(V(G)\times V(H)\) with edges of two types:\N\begin{itemize}\N\item type-1 edge: \((g,h)(g,h^\prime)\) is an edge of \(G\otimes_f H\) for every vertex \(g\in V(G)\) and every edge \(hh^\prime\in E(H)\),\N\item type-2 edge: \((g,f(g^\prime))(g^\prime,f(g))\) is an edge of \(G\otimes_f H\) for every edge \(gg^\prime\in E(G)\).\N\end{itemize}\NLet \(H^G\) be the family of functions from \(V(G)\) to \(V(H)\). The authors introduce new types of domination, the Sierpiński domination number, denoted by \(\gamma_S(G,H)\), as the minimum over all functions \(f\) from \(H^G\) of the domination number of the Sierpiński product with respect to \(f\), and upper Sierpiński domination number, denoted by \(\Gamma_S(G,H)\), as the maximum over all functions \(f\in H^G\) of domination number of the Sierpiński product with respect to \(f\). The authors after establishing general upper and lower bounds, determine the upper Sierpiński domination number of the Sierpiński product of two cycles and determine the lower Sierpiński domination number of the Sierpiński product of two cycles in half of the cases and in the other half cases restrict it to two values.