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

TL;DR

This paper investigates the resolvability and convexity properties of the Sierpiński product of graphs, providing formulas for metric dimensions and analyzing convex layers, especially for trees and certain cycles.

Contribution

It introduces formulas for Sierpiński metric dimensions, characterizes convex layers, and explores properties for specific graph classes like trees and cycles.

Findings

Determined Sierpiński metric and upper metric dimensions.

Provided closed formulas for trees and cycles with triangles.

Proved convexity of layers in Sierpiński product 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.