Resolvability and convexity properties in the Sierpi\'{n}ski product of graphs

Abstract

Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpi\'{n}ski 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 $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. The Sierpi\'{n}ski metric dimension and the upper Sierpi\'{n}ski metric dimension of two graphs are determined. Closed formulas are determined for Sierpi\'{n}ski products of trees, and for Sierpi\'{n}ski products of two cycles where the second factor is a triangle. We also prove that the layers with respect to the second factor in a Sierpi\'{n}ski product graph are convex.