Global r-alliances and total domination

• Characterized by the vertices in their neighborhood outside of the alliance,written as ∂OA := N[OA] \ OA.• Every such vertex has at least as many bonds to members in the alliancethan to non-members (including itself).• An offensive alliance can attack every neighbor successfully.• graph-theoretic notation: OA ⊆ V , such that for every v ∈ ∂OA: |N