In this work, we analyze a sequential game played in a graph called the Multilevel Critical Node problem (MCN). A defender and an attacker are the players of this game. The defender starts by preventively interdicting vertices (vaccination) from being attacked. Then, the attacker infects a subset of non-vaccinated vertices and, finally, the defender reacts with a protection strategy. We provide the first computational complexity results associated with MCN and its subgames. Moreover, by considering unitary, weighted, undirected, and directed graphs, we clarify how the theoretical tractability of those problems vary. Our findings contribute with new NP-complete, $\Sigma_2^p$-complete and $\Sigma_3^p$-complete problems. Furthermore, for the last level of the game, the protection stage, we build polynomial time algorithms for certain graph classes.

中文翻译：

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多级关键节点问题的复杂性

在这项工作中，我们分析了在称为多级关键节点问题 (MCN) 的图中进行的顺序游戏。防守者和进攻者是这个游戏的参与者。防御者首先预防性地阻止顶点（接种疫苗）受到攻击。然后，攻击者感染一部分未接种疫苗的顶点，最后，防御者以保护策略作出反应。我们提供了与 MCN 及其子博弈相关的第一个计算复杂度结果。此外，通过考虑酉图、加权图、无向图和有向图，我们阐明了这些问题的理论可处理性如何变化。我们的发现有助于解决新的 NP-complete、$\Sigma_2^p$-complete 和 $\Sigma_3^p$-complete 问题。此外，对于游戏的最后一个级别，即保护阶段，我们为某些图类构建多项式时间算法。