The Critical Node Problem Based on Connectivity Index and Properties of Components on Trees

摘要

We deal with the critical node problem (CNP) in a graph G, in which a given number K of nodes are removed to minimize the connectivity of the residual graph in some sense. Several ways to minimize some connectivity measurement have been proposed, including minimizing the connectivity index(MinCI), maximizing the number of components, minimizing the maximal component size. We propose two classes of CNPs by combining the above measurements together. The objective is to minimize the sum of connectivity indexes and the total degrees in the residual graph. The CNP with an upper-bound M on the maximal component size is denoted by MSCID-CS and the one with an extra upper-bound P on the number of components is denoted by MSCID-CSN. They are generalizations of the MinCI, which has been shown NP-hard for general graphs. In particular, we study the case where G is a tree. Two dynamic programming algorithms are proposed to solve the two classes of CNPs. The time complexities of the algorithms for MSCID-CS and MSCID-CSN are O(n2K2M2) and O(n2K2M2P2), respectively, where n is the number of nodes in G. Computational experiments are presented which show the effectiveness of the algorithms.