Feedback vertex number of Sierpiński-type graphs

Abstract

The feedback vertex number $τ(G)$ 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 $τ(S_p^n)=p^{n-1}(p-2)$ for the Sierpiński graph $S_p^n$ with $p\geq 2$ and $n\geq 1$. The generalized Sierpiński triangle graph $\hat{S_p^n}$ is obtained by contracting all non-clique edges from the Sierpiński graph $S_p^{n+1}$. We prove that $τ(\hat{S}_3^n)=\frac {3^n+1} 2=\frac{|V(\hat{S}_3^n)|} 3$, and give an upper bound for $τ(\hat{S}_p^n)$ for the case when $p\geq 4$.