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 \(\ell \leq k\) vertices from \(X\) followed by the addition of any \(\ell \sim 1\) vertices of G
results in a set which does not dominate \(G\). The \(k\)-minimal domination number IWRC)
of \(G\) is the largest number of vertices in a k-minimal dominating set of G. The sequence
\(R:m_1 \geq m_2 \geq… \geq m_k \geq …. \geq\) n of positive integers is a domination sequence iff
there exists a graph \(G\) such that \(\Gamma_1 (G) = m_1, \Gamma_2(G) = m_2,… \Gamma_k(G) = m_k,…,\)
and \(\gamma(G) = n\), where \(\gamma(G)\) denotes the domination number of G. We give sufficient
conditions for R to be a domination sequence.