(experimental page)

The paper by William F. Klostermeyer, Margaret-Ellen Messinger, Alejandro Angeli Ayello study the $DD_m(G)$ and its connection to various graph parameters.

Definitions

• independence number $\alpha(G)$
• dominating set
• domination number $\gamma(G)$
• perfect matching
• eternal eviction model
• (dominating) swap set
• $DD_m(G)$
• weak & strong stem
• weak & strong graph
• (simple) star partitioning
• weight of a partitioning
• $\widehat{G}$ graph
• weak reduction

Results

• For every tree $\gamma(T) = e^\infty_m(T) = DD_m(T) = \alpha(T)$ if and only if $T = \widehat{H}$ for some tree $H$ of order $2|V(T)|$.
• For any tree $T, e^\infty_m(T) = \alpha(T)$ if and only if T has a minimum-weight simple star partitioning containing no $K_1$ parts.
• For $T’$ a week reduction of T, $S(T’) = \frac{1}{2} |V(T)| ⇔ \alpha(T) = S(T)$
• $\alpha(T) = e^\infty_m(T) \Leftrightarrow S(T) = 2|V(T)|$
• Let $p, q \geq 1$. Then $DD_m(K_{1,p} ~\square~ K_{1,q}) = p+q-1$.