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Binary arc variables with one outgoing and one incoming arc at each vertex describe a cycle cover of a directed graph. For every nonempty proper vertex subset $S$, the displayed inequality forbids a directed cycle confined to $S$. Together these inequalities make the cycle cover one Hamiltonian cycle. They can be generated as violated cuts or used in branch and bound, rather than all being stored initially.

For overlapping solid bodies, the penetration depth is the smallest translation needed to remove interior overlap, with orientation fixed. It is distance from the origin to the complement of the translational collision region. Define signed separation as positive ordinary separation for disjoint bodies, zero for contact, and $-\delta$ for interior overlap. Translation-space branch and bound can bracket it using certified intersection/containment queries. This convention handles crossing and complete containment; merely negating unsigned boundary distance does not.