Smallest Domination Number and Largest Independence Number of Graphs and Forests with given Degree Sequence

For a sequence $d$ of non-negative integers, let ${\cal G}(d)$ and ${\cal F}(d)$ be the sets of all graphs and forests with degree sequence $d$, respectively. Let $γ_{\min}(d)=\min\{ γ(G):G\in {\cal G}(d)\}$, $α_{\max}(d)=\max\{ α(G):G\in {\cal G}(d)\}$, $γ_{\min}^{\cal F}(d)=\min\{ γ(F):F\in {\cal F}(d)\}$, and $α_{\max}^{\cal F}(d)=\max\{ α(F):F\in {\cal F}(d)\}$ where $γ(G)$ is the domination number and $α(G)$ is the independence number of a graph $G$. Adapting results of Havel and Hakimi, Rao showed in 1979 that $α_{\max}(d)$ can be determined in polynomial time. We establish the existence of realizations $G\in {\cal G}(d)$ with $γ_{\min}(d)=γ(G)$, and $F_γ,F_α\in {\cal F}(d)$ with $γ_{\min}^{\cal F}(d)=γ(F_γ)$ and $α_{\max}^{\cal F}(d)=α(F_α)$ that have strong structural properties. This leads to an efficient algorithm to determine $γ_{\min}(d)$ for every given degree sequence $d$ with bounded entries as well as closed formulas for $γ_{\min}^{\cal F}(d)$ and $α_{\max}^{\cal F}(d)$.

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