Equality of Maker–Breaker domination parameters for random graphs

Let $G$ be a random graph, and let $\gamma_{MB}(G,b)$ and $s_{MB}(G,b)$ denote the two Maker–Breaker domination game parameters. For a Binomial random graph $G_{n,p}$ with constant edge probability $p\in (0,1)$, or a random geometric graph $\mathcal{G}_2(n,r)$ with constant radius $r\in (0,1)$, the random-graph equality conjecture.

$$\gamma_{MB}(G,b)=s_{MB}(G,b)$$

holds with high probability.

The conjecture predicts that these two game parameters coincide asymptotically almost surely in the stated random-graph models. The surrounding discussion notes that related Maker–Breaker domination parameters asymptotically equal the domination number with high probability; the equality with $s_{MB}$ remains conjectural.