Resolvability and convexity properties in the Sierpiński product of graphs

Henning, Michael A. (author), Klavžar, Sandi (author), Yero, Ismael G. (author)

Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ 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 $gg'$ 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 Sierpiński metric dimension and the upper Sierpiński metric dimension of two graphs are determined. Closed formulas are determined for Sierpiński products of trees, and for Sierpiński 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 Sierpiński product graph are convex.

• Language: English
• Keywords: Sierpiński product of graphs, metric dimension, trees, convex subgraph
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 17 str.
• Numbering: Vol. 21, iss. 1, art. 3
• PID: 20.500.12556/RUL-152589
• UDC: 519.17
• ISSN on article: 1660-5446
• DOI: 10.1007/s00009-023-02544-6
• COBISS.SI-ID: 172901379
• Publication date in RUL: 29.11.2023

Record is a part of a journal

• Title: Mediterranean journal of mathematics
• Shortened title: Mediterr. j. math.
• Publisher: Springer Nature, Birkhäuser
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: Sierpinskijev produkt grafov, metrična dimenzija, drevesa, konveksni podgrafi