Bounds for scattering number and rupture degree of graphs with genus

For a given graph G = (V, E), denote by m(G) and omega(G) the order of the largest component and the number of components of G, respectively. The scattering number of G is defined as s(G) = max{omega(G - X) - vertical bar X vertical bar : X subset of V, omega(G - X) > 1}, and the rupture degree r(G) = max{omega(G - X) - vertical bar X vertical bar - m(G - X) : X subset of V(G), omega(G - X) > 1}. These two parameters are related to reliability and vulnerability of networks. In this paper, we present some new bounds on the scattering number and rupture degree of a graph G in terms of its connectivity kappa(G) and genus gamma(G). Furthermore, we give graphs to show these bounds are best possible. (C) 2018 Elsevier Inc. All rights reserved.