On Critical Node Problems with Vulnerable Vertices

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.

We perform a parameterized complexity study of both problems. For example, we show that both problems are FPT with respect to $k+x$. Furthermore, in case of deletable vulnerable vertices we provide a polynomial kernel for the parameter ${{\,\mathrm{vc}\,}}+k$, where ${{\,\mathrm{vc}\,}}$ is the vertex cover number. In case of non-deletable vulnerable vertices, we prove NP-hardness even when there is only one vulnerable vertex.

Most of the results of this work are also contained in the first author’s Master’s thesis [12].

F. Sommer—Supported by the DFG, project EAGR KO 3669/6-1.