Vertex-connectivity of minimal Maker–Breaker graphs for cliques

Determine the smallest vertex-connectivity among all graphs in \mathcal{M}_{MB}^1(K_t).

Background

The paper compares minimal Maker–Breaker graphs with minimal Ramsey graphs, for which the smallest vertex-connectivity of a minimal Ramsey graph for K_t is known to be 3. The corresponding parameter for minimal unbiased Maker–Breaker-winnable graphs remains unresolved.

References

Determine the smallest vertex-connectivity among all graphs in \mathcal{M}1_{MB}(K_t).

— Ramsey-type results for Maker-Breaker games  (2609.02588 - Allin et al., 2 Sep 2026) in Problem following Theorem 5.1, Concluding remarks and open problems

Given t\in \mathbb{N}. Characterize all connected graphs H such that H K_t.

— Ramsey-type results for Maker-Breaker games  (2609.02588 - Allin et al., 2 Sep 2026) in Problem following Theorem 5.2, subsection “Equivalent graphs for cliques,” Concluding remarks and open problems