Best upper bound for generalized power domination in regular graphs

Determine the best possible upper bound for the k-power domination number \(\gamma_{P,k}(G)\) of a connected \(r\)-regular graph when \(1\leq k\leq r-3\), without assuming that the graph is claw-free.

Background

The paper studies the k-power domination number of regular graphs. It proves the conjectured sharp bound for connected claw-free $(k+\ell+1)$-regular graphs when $k\geq\ell$, but removes the claw-free assumption only asymptotically for fixed $k$.

For general connected $r$-regular graphs, earlier conjectures predicting an upper bound of $n/(r+1)$ were disproved in the range $1\leq k\leq r-3$. The paper determines the asymptotic behavior of the supremal ratio $\gamma_{P,k}(G)/|V(G)|$ as $(\ln r)/r$ for fixed $k$, but does not determine the exact best possible upper bound in this range.