发表机构
Izmir Bakircay University（伊兹密尔巴基尔查伊大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究单节点攻击下三种关键性目标（PC、LCC、NC）间的目标转移损失，给出PC/LCC间最优常数界，证明NC相关方向无正常数，并确定最优顶点集分离的最小图阶数。
AI 中文摘要
当一个顶点在一种破坏目标下被选为最具破坏性的单一攻击目标时，它在另一种目标下能保留多少最优破坏？我们针对关键节点检测中常用的三种目标——成对连通性（PC）、最大幸存组件规模（LCC）和组件数量（NC）——研究了连通图上单顶点攻击的这一问题，其中被攻击顶点计入破坏，平局按乐观方式解决。对于PC和LCC这对目标，损失是有界的：PC最优顶点至少保留最优LCC破坏的$2-\sqrt2$比例，LCC最优顶点至少保留最优PC破坏的2/3，且这两个常数都是最优可能的。这两个界都不依赖于平局最优顶点的选择，也不依赖于被攻击顶点是否被计入，并且作为任何固定大小攻击集的下界仍然有效。对于涉及NC的四个方向，不存在正的常数；显式构造族的比例按1/k衰减，且没有任何单一攻击集能统一保留所有三个最优目标的正比例。我们还确定了最优顶点集变得不相交的最小阶数：LCC/NC为7，PC/NC为8，PC/LCC为9，三者两两分离为11，并且对于每个n≥11，树都能实现三重分离；这些值基于对3≤n≤10的11,989,762个连通图的穷举枚举。最后，在Birnbaum意义上最关键的顶点可能依赖于失效概率：一个六顶点图（最小可能阶数）在$p^*=3-\sqrt5$处改变一次领导者。添加通用顶点将预算为一的示例提升到每个固定预算，因此在所有连通图上，相同的常数在每个固定预算下都是最优的；在树等受限类中的行为留待进一步研究。
英文摘要
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.
Comments20 pages, 2 figures, 1 table. Ancillary files: replay scripts and exact certificates (anc/)
