On Isolated Rupture Degree of Graphs

The isolated rupture degree for a connected graph G is defined as ir(G) = max{i(G - S) - vertical bar S vertical bar - m(G - S) : S (sic) C(G)}, where i(G - S) and m(G - S), respectively, denote the number of components which are isolated vertices and the order of a largest component in G - S. C(G) denotes the set of all cut-sets of G. The isolated rupture degree is a new graph parameter which can be used to measure the vulnerability of networks. In this paper, we give isolated rupture degrees of several specific classes of graphs. Formulas for the isolated rupture degree of join graphs and some bounds of the rupture degree are given. We also determine the isolated rupture degree of grids, and that of the hypercubes as a special case.