Improved upper bounds for Staller–Maker–Breaker domination numbers

Let $G$ be a graph on $n$ vertices, with minimum degree $\delta(G)$. The S-game and D-game parameters are denoted by $\gamma_{\rm SMB}'(G)$ and $\gamma_{\rm SMB}(G)$, respectively.

Improved SMBD-number bounds.

$$\begin{aligned} &\text{(i) If }\gamma_{\rm SMB}'(G)<\infty,\text{ then }\gamma_{\rm SMB}'(G)\le \left\lceil\frac{n}{2}\right\rceil-\delta(G)+1;\\ &\text{(ii) If }\gamma_{\rm SMB}(G)<\infty,\text{ then }\gamma_{\rm SMB}(G)\le \left\lfloor\frac{n}{2}\right\rfloor-\delta(G)+1. \end{aligned}$$

These bounds would improve the known sharp general upper bounds when the minimum degree is at least two; the authors note that sharp examples under this degree condition are not known.