Domination Sequences of Graphs

Abstract

A dominating set $X$ of a graph $G$ is a k-minimal dominating set of $G$ iff the
removal of any $\ell \le k$ 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
$R:m_1\ge m_2\ge \dots \ge m_k\ge \dots .\ge$ n of positive integers is a domination sequence iff
there exists a graph $G$ such that ${\mathrm{\Gamma }}_1(G)=m_1,{\mathrm{\Gamma }}_2(G)=m_2,\dots {\mathrm{\Gamma }}_k(G)=m_k,\dots ,$
and $\gamma (G)=n$, where $\gamma (G)$ denotes the domination number of G. We give sufficient
conditions for R to be a domination sequence.