DOMINATION SEQUENCES OF GRAPHS

A dominating set X of a graph G is a k -minimal dominating set of G iff the removal of any l less-than-or-equal-to k vertices from X followed by the addition of any l-1 vertices of G results in a set which does not dominate G. The k-minimal domination number LAMBDA(k)(G) of G is the largest number of vertices in a k-minimal dominating set of G. The sequence R: m1 greater-than-or-equal-to m2 greater-than-or-equal-to ... greater-than-or-equal-to m(k) greater-than-or-equal-to ... greater-than-or-equal-to n of positive integers is a domination sequence iff there exists a graph G such that LAMBDA-1(G) = m1, LAMBDA-2(G) = m2,..., LAMBDA(k)(G) = m(k),..., and gamma(G) = n, where gamma(G) denotes the domination number of G. We give sufficient conditions for R to be a domination sequence.