The one-half Minimizer-start Enclaveless Game Conjecture

Let $G$ be a graph of order $n$, and let $\delta(G)$ denote its minimum degree. In the Minimizer-start competition-enclaveless game, Minimizer starts, and $\Psi_g^-(G)$ is the number of vertices chosen when both players use optimal strategies. The one-half Minimizer-start Enclaveless Game Conjecture.

$$\delta(G) \ge 2 \quad\Longrightarrow\quad \Psi_g^-(G) \ge \frac{1}{2}n.$$

This conjecture gives a corresponding lower bound for the Minimizer-start version of the competition-enclaveless game. The source presents it as one of the main unsettled questions motivating the paper.