arXiv:2610.37239v1 Announce Type: cross Abstract: Multi-relational graph detectors learn from clean data how much to trust each relation, but an adversary pays a different price to manipulate each relation and each action; taking such costs as given, we ask when using a relation is worse than ignoring it. Our first main result proves that degree gating strictly beats every selection or reweighting of a relation when part of the population is isolated on it and the attacker cannot afford to open the gate, and that this window is sharp; on real graphs, validated gating gains exactly on the relations that meet these conditions and nowhere else. Our second shows that a per-node exposure bound, which extends to message passing, tells a platform which relation to harden: exposure per unit cost, computed without any attack, picks the relation whose hardening raises the exact trap threshold most in 15 of 16 cases, against at most 7 for informativeness, cheapness, or an analysis with equal edit costs. Supporting tools give exact worst cases for linear relational models, certified brackets for R-GCN and tree ensembles, and a finite-sample guarantee for adversarial validation, under which clean-trained graph detectors fall below their graph-free counterparts in 58 of 90 settings on six graphs. A selection protocol built on these results is never significantly worse than ignoring the graph; with at most four relations it mostly abstains, as validation alone would, and its cost-aware proposals help when relations are many, so far shown on synthetic graphs. Without cost information, the safe choice is to abstain.