Graphs with given connectivity and independence number or networks with given measures of vulnerability and survivability

The criterion for invulnerability of a network based on the connectivity of a graph is well treated in literature. We define a criterion for survivability of a network in terms of the independence number of a graph. The following problems are then considered. 1) Find anr-connected graph, withnvertices andm_0 = [(nr + l)/2]edges, whose independence number is the minimum possible, where[x]denotes the greatest integer less than or equal tox cdot 2) Given positive integersn,m,r, andk, find the realizability conditions for a graph withnvertices andmedges, whose connectivity is at leastrand independence number is at mostk.