A polynomial algorithm for computing the weak rupture degree of trees

Let G = (V, E) be a graph. The weak rupture degree of G is defined as r(w) (G) = max{omega(G -X - vertical bar X vertical bar - m(e) (G -X) : omega(G - X) > 1}, where the maximum is taken over all X, the subset of V(G), omega(G - X) is the number of components in G - X, and m(e) (G - X) is the size (edge number) of a largest component in G - X. This is an important parameter to quantitatively describe the invulnerability of networks. In this paper, based on a study of relationship between network structure and the weak rupture degree, a polynomial algorithm for computing the weak rupture degree of trees is given. (C) 2019 Elsevier Inc. All rights reserved.