On Critical Node Problems with Vulnerable Vertices

Jannik Schestag, Niels Grüttemeier, Christian Komusiewicz, Frank Sommer

Abstract: A vertex pair in an undirected graph is called connected if the two vertices are in the same connected component. In the NP-hard Critical Node Problem (CNP), the input is an undirected graph G with integers k and x, and the question is whether we can transform G via at most k vertex deletions into a graph whose total number of connected vertex pairs is at most x. In this work, we introduce and study two NP-hard variants of CNP where a subset of the vertices is marked as vulnerable and we aim to obtain a graph with at most x connected vertex pairs where at least one vertex is vulnerable. In the first variant, which generalizes CNP, we may delete vulnerable and non-vulnerable vertices. In the second variant, we may only delete non-vulnerable vertices.