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 rw(G)=max{ω(G−X)−|X|−me(G−X):ω(G−X)>1}, where the maximum is taken over all X, the subset of V(G), ω(G−X) is the number of components in G−X, and me(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.