Characterization of the upper-extremal case when the minimum degree is at most k

Characterize the graphs G satisfying γ_k^L(G)=n-k when the degree parameters satisfy δ(G)≤k<Δ(G).

Background

The general upper bound γ_kL(G)≤n-k is attained by several graph families. The paper provides a characterization under the assumption k<δ(G), including a complete characterization for regular graphs.

The structure of graphs attaining γ_kL(G)=n-k remains unresolved in the intermediate degree range where the minimum degree is at most k but k is still smaller than the maximum degree. The open problem asks for a characterization in precisely this omitted parameter regime.

References

While the previous question concerns the smallest possible value of $\gamma_kL(G)$, it is also natural to investigate the opposite extreme case. In this paper, we characterize graphs satisfying $\gamma_kL(G)=n-k$ under the assumption $k< \delta(G)$. However, the structure of such graphs remains unknown when $\delta(G)\le k<\Delta(G)$.

Can we characterize the graphs $G$ satisfying $\gamma_kL(G)=n-k$ when $\delta(G)\le k<\Delta(G)$?

— 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 discussion of the case k<δ(G)