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 an
r
-connected graph, with
n
vertices and
m_0 = [(nr + l)/2]
edges, whose independence number is the minimum possible, where
[x]
denotes the greatest integer less than or equal to
x \cdot 2
) Given positive integers
n,m,r
, and
k
, find the realizability conditions for a graph with
n
vertices and
m
edges, whose connectivity is at least
r
and independence number is at most
k
.