The domatic number of a graph is the maximum number of disjoint dominating sets that partition the vertices — the maximum number of “shifts” that can independently cover the entire graph. The game domatic number is the same concept under adversarial conditions: two players alternate assigning vertices to dominating sets, with one player maximizing the number of valid partitions and the other minimizing it.
English and Swan construct graphs where the game domatic number is 1 despite arbitrarily high classical domatic number. The adversary can prevent even two valid dominating sets from forming, regardless of how many are possible under cooperative play.
The construction exploits the sequentiality of the game. The adversary uses early moves to create structural commitments that force the maximizer into impossible positions later. The key is that domination requires covering ALL vertices — a single uncovered vertex invalidates the entire set. The adversary targets vertices that are hard to dominate from the remaining unassigned vertices, creating a cascading failure.
The separation is stark: domatic number k (for any k) versus game domatic number 1. The gap between cooperative and adversarial partitioning is not bounded by any function of the cooperative optimum. No amount of cooperative potential protects against adversarial play.
This contrasts with many graph parameters where the game version is within a constant factor of the classical version. Domination's all-or-nothing coverage requirement creates a vulnerability that adversaries exploit — the failure mode is not gradual degradation but complete collapse.