General k-limited domination number of Sierpiński graphs

Determine whether the k-limited domination number of the Sierpiński graph S(n,m) satisfies \(\gamma_k^L(S(n,m))=(m-k)m^{n-1}\) for all integers \(n\ge 1\), \(m\ge 2\), and \(1\le k\le m-1\), thereby completely determining this parameter for Sierpiński graphs.

Background

The paper establishes the exact value $\gamma_1^L(S(n,m))=(m-1)m^{n-1}$ for all Sierpiński graphs and determines $\gamma_k^L(S(n,3))=(3-k)3^{n-1}$ for $k\in\{1,2\}$. These results suggest the broader formula $\gamma_k^L(S(n,m))=(m-k)m^{n-1}$ throughout the full parameter range $1\le k\le m-1$.

Proving the conjecture would extend the paper's exact calculations from the cases $k=1$ for arbitrary $m$ and $k\in\{1,2\}$ for $m=3$ to every admissible value of $k$ and $m$.