Characterization of graphs attaining the domination-number lower bound

Characterize the graphs G for which the k-limited domination number satisfies γ_k^L(G)=γ(G).

Background

For every graph in the considered setting, the k-limited domination number γ_kL(G) is bounded below by the ordinary domination number γ(G) and by ⌈n/(k+1)⌉. The paper characterizes graphs attaining the latter bound, but leaves unresolved the characterization of graphs attaining the former bound.

Previous work showed that equality γ_kL(G)=γ(G) can occur, including for efficient graphs under additional assumptions. The unresolved problem asks for a complete characterization of all graphs for which the capacity constraint does not increase the minimum domination number.

References

Several natural questions remain open. The parameter $\gamma_kL(G)$ is naturally bounded below by both $\gamma(G)$ and $\lceil\frac{n}{k+1}\rceil$. While one of the main results of this paper provides a characterization of graphs attaining the bound $\gamma_kL(G)=\lceil\frac{n}{k+1}\rceil$, the question of when the other lower bound is attained remains open. In , it was shown that the equality $\gamma_kL(G)=\gamma(G)$ can occur, in particular for efficient graphs under some additional assumption. This naturally leads to the following question.

Can we characterize the graphs $G$ for which $\gamma_kL(G)=\gamma(G)$?

— An improved lower and upper bound of the k-limited domination number  (2610.01442 - Radić, 1 Oct 2026) in Section 5, Conclusion; Open problem immediately following the first paragraph of open questions