Complexity of the multilevel critical node problem

In this work, we analyze a sequential game played in graph called the Multilevel Critical Node problem (MCN). A defender and an attacker are players of game. The starts by preventively interdicting vertices (vaccination) from being attacked. Then, infects subset non-vaccinated and, finally, reacts with protection strategy. We provide first computational complexity results associated MCN its subgames. Moreover, considering unitary, weighted, undirected, directed graphs, clarify how theoretical tractability those problems vary. Our findings contribute new NP-complete, Σ2p-complete Σ3p-complete problems. Furthermore, for last level game, stage, build polynomial time algorithms certain classes.