Mutual-visibility and general position sets in Sierpi\'nski triangle graphs

TL;DR

This paper investigates the maximum size of special vertex sets in Sierpiński triangle graphs, focusing on the general position and mutual-visibility problems, which relate to shortest paths and visibility between vertices.

Contribution

It provides new results on the general position and mutual-visibility numbers specifically for Sierpiński triangle graphs, expanding understanding of these concepts in fractal-like structures.

Findings

Determined the mutual-visibility number for Sierpiński triangle graphs.

Established bounds for the general position number in these graphs.

Analyzed variations like total, outer, and dual mutual-visibility sets.

Abstract

For a given graph \(G\), the general position problem asks for the largest set of vertices \(M \subseteq V(G)\) such that no three distinct vertices of \(M\) belong to a common shortest path in \(G\). A relaxation of this concept is based on the condition that two vertices \(x, y \in V(G)\) are \(M\)-visible, meaning there exists a shortest \(x, y\)-path in \(G\) that does not pass through any vertex of \(M \setminus \{x, y\}\). If every pair of vertices in \(M\) is \(M\)-visible, then \(M\) is called a mutual-visibility set of \(G\). The size of the largest mutual-visibility set of \(G\) is called the mutual-visibility number of \(G\). Some well-known variations of this concept consider the total, outer, and dual mutual-visibility sets of a graph. We present results on the general position problem and the various mutual-visibility problems in Sierpi\'nski triangle graphs.