Feedback vertex number of Sierpi\'{n}ski-type graphs

TL;DR

This paper determines the feedback vertex number for Sierpiński-type graphs, providing exact formulas for certain cases and bounds for others, advancing understanding of cycle removal in fractal-like graphs.

Contribution

It derives explicit formulas for the feedback vertex number of Sierpiński graphs and bounds for generalized Sierpiński triangle graphs, extending known results in graph theory.

Findings

Exact formula for $ au(S_p^n)$ when $p eq 2$

Exact feedback vertex number for $ au( ilde{S}_3^n)$

Upper bounds for $ au( ilde{S}_p^n)$ when $p eq 3$

Abstract

The feedback vertex number $\tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $\tau(S_p^n)=p^{n-1}(p-2)$ for the Sierpi\'{n}ski graph $S_p^n$ with $p\geq 2$ and $n\geq 1$. The generalized Sierpi\'{n}ski triangle graph $\hat{S_p^n}$ is obtained by contracting all non-clique edges from the Sierpi\'{n}ski graph $S_p^{n+1}$. We prove that $\tau(\hat{S}_3^n)=\frac {3^n+1} 2=\frac{|V(\hat{S}_3^n)|} 3$, and give an upper bound for $\tau(\hat{S}_p^n)$ for the case when $p\geq 4$.