The Sierpiński Domination Number

Abstract

Let $G$ and $H$ be graphs and let $f:V(G)\to V(H)$ be a function. The Sierpiński product of $G$ and $H$ with respect to $f$, denoted by $G{\otimes }_{f}H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $g{g}^{\prime }$ of $G$ there is an edge between copies $gH$ and $g^{\prime }H$ of $H$ associated with the vertices $g$ and $g^{\prime }$ of $G$, respectively, of the form $(g,f(g^{\prime }))(g^{\prime },f(g))$. In this paper, we define the Sierpiński domination number as the minimum of $\gamma (G{\otimes }_{f}H)$ over all functions $f:V(G)\to V(H)$. The upper Sierpiński domination number is defined analogously as the corresponding maximum. After establishing general upper and lower bounds, we 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.