Varieties of mutual-visibility and general position on Sierpiński graphs

TL;DR

This work investigates eight invariants related to mutual-visibility and general-position on Sierpiński graphs $S_p^n$, focusing on exact values for $S_p^2$ with all $p\\ge 3$ and extensive results for $S_3^n$ (often with complete extremal-set enumerations). The authors derive precise formulas for $\,\\mu(S_p^2)$ and $gp(S_p^2)$, along with the dual/outer variants, and then establish deep inductive results for $S_3^n$, including $\\mu(S_3^n)=gp(S_3^n)=3^{n-2}+3$ for $n\\ge 2$, and fixed values $\\mu_t(S_3^n)=\\mu_d(S_3^n)=gp_t(S_3^n)=gp_d(S_3^n)=3$ with explicit extremal constructions. Outer-invariant values remain open in the $S_3^n$ case, while the paper provides tight structural bounds and detailed small-n enumerations that guide the general proofs. The results connect to broader themes in graph products and contractions of Sierpiński graphs and contribute exact, enumerated extremal configurations useful for both theoretical and algorithmic perspectives on visibility-like properties in fractal-like graphs.

Abstract

The variety of mutual-visibility problems contains four members, as does the variety of general position problems. The basic problem is to determine the cardinality of the largest such sets. In this paper, these eight invariants are investigated on Sierpiński graphs $S_p^n$. They are determined for the Sierpiński graphs $S_p^2$, $p\ge 3$. All, but the outer mutual-visibility number and the outer general position number, are also determined for $S_3^n$, $n\ge 3$. In many of the cases the corresponding extremal sets are enumerated.

Figures (3)

• Figure 1: $S_3^4$ and some of its bulls
• Figure 2: $\mu$-sets in $S_3^2$ and in $S_4^2$
• Figure 3: $\mu_{\rm d}$-sets in $S_4^2$

Theorems & Definitions (15)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Corollary 3.4
• proof
• Theorem 3.5