  Let G be a simple graph with vertex set V(G). A subset S of V(G) is
independent if no two vertices from S are adjacent. The graph G is known to be
a Konig-Egervary if alpha(G) + mu(G)= |V(G)|, where alpha(G) denotes the size
of a maximum independent set and mu(G) is the cardinality of a maximum
matching. Let Omega(G) denote the family of all maximum independent sets, and f
be the function from the set of subcollections Gamma of Omega(G) such that
f(Gamma) = (the cardinality of the union of elements of Gamma) + (the
cardinality of the intersection of elements of Gamma). Our main finding claims
that f is "<<"-increasing, where the preorder {Gamma1} << {Gamma2} means that
the union of all elements of {Gamma1} is a subset of the union of all elements
of {Gamma2}, while the intersection of all elements of {Gamma2} is a subset of
the intersection of all elements of {Gamma1}. Let us say that a family {Gamma}
is a Konig-Egervary collection if f(Gamma) = 2*alpha(G). We conclude with the
observation that for every graph G each subcollection of a Konig-Egervary
collection is Konig-Egervary as well.
