arXiv2026
When a vertex is selected as the most damaging single attack target under one damage objective, how much of the optimal damage under a different objective does it retain? We study this for the three objectives commonly used in critical node detection -- pairwise connectivity (PC), size of the largest surviving component (LCC) and number of components (NC) -- for single-vertex attacks on connected graphs, with the attacked vertex counted in the damage and ties resolved optimistically. For the pair PC, LCC the loss is bounded: a PC-optimal vertex retains at least a $2-\sqrt2$ fraction of the optimal LCC damage, an LCC-optimal vertex at least 2/3 of the optimal PC damage, and both constants are best possible. Neither bound depends on the choice among tied optimal vertices or on whether the attacked vertex is counted, and both remain valid as lower bounds for attack sets of any fixed size. For the four directions involving NC no positive constant exists; explicit families have ratios decaying like 1/k, and no single attack set retains a positive fraction of all three optima uniformly. We also determine the smallest orders at which the sets of optimal vertices become disjoint: 7 for LCC/NC, 8 for PC/NC, 9 for PC/LCC and 11 for all three pairwise, with trees realising the triple separation for every n>=11; these values rest on an exhaustive enumeration of the 11,989,762 connected graphs with 3<=n<=10. Finally, the most critical vertex in a Birnbaum-type sense may depend on the failure probability: a six-vertex graph, of smallest possible order, changes leader once, at $p^*=3-\sqrt5$. Adding universal vertices lifts the budget-one examples to every fixed budget, so over all connected graphs the same constants are optimal at every fixed budget; behaviour within restricted classes such as trees is left open.