A polynomial algorithm for computing the weak rupture degree of trees

Abstract Let G = ( V , E ) be a graph. The weak rupture degree of G is defined as r w ( G ) = max { ω ( G − X ) − | X | − m e ( 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 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.