Bounds for scattering number and rupture degree of graphs with genus

For a given graph G=(V,E), denote by m(G) and ω(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{ω(G−X)−|X|:X⊆V,ω(G−X)>1}, and the rupture degree r(G)=max{ω(G−X)−|X|−m(G−X):X⊆V(G),ω(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 κ(G) and genus γ(G). Furthermore, we give graphs to show these bounds are best possible.