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

TL;DR

This paper investigates extremal properties of graphs and forests with a given degree sequence, establishing existence of specific realizations and providing efficient algorithms and formulas for key graph invariants.

Contribution

It proves the existence of realizations with extremal domination and independence numbers and offers algorithms and formulas for these parameters.

Findings

Existence of realizations with extremal properties.

Efficient algorithm for minimum domination number with bounded degree sequences.

Closed-form formulas for maximum independence and minimum domination in forests.

Abstract

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 $\gamma_{\min}(d)=\min\{ \gamma(G):G\in {\cal G}(d)\}$, $\alpha_{\max}(d)=\max\{ \alpha(G):G\in {\cal G}(d)\}$, $\gamma_{\min}^{\cal F}(d)=\min\{ \gamma(F):F\in {\cal F}(d)\}$, and $\alpha_{\max}^{\cal F}(d)=\max\{ \alpha(F):F\in {\cal F}(d)\}$ where $\gamma(G)$ is the domination number and $\alpha(G)$ is the independence number of a graph $G$. Adapting results of Havel and Hakimi, Rao showed in 1979 that $\alpha_{\max}(d)$ can be determined in polynomial time. We establish the existence of realizations $G\in {\cal G}(d)$ with $\gamma_{\min}(d)=\gamma(G)$, and $F_{\gamma},F_{\alpha}\in {\cal F}(d)$ with $\gamma_{\min}^{\cal F}(d)=\gamma(F_{\gamma})$ and $\alpha_{\max}^{\cal F}(d)=\alpha(F_{\alpha})$ that have strong structural properties. This leads to an efficient algorithm to determine $\gamma_{\min}(d)$ for every given degree sequence $d$ with bounded entries as well as closed formulas for $\gamma_{\min}^{\cal F}(d)$ and $\alpha_{\max}^{\cal F}(d)$.