Some Comments on the Slater number

TL;DR

This paper investigates the Slater number as a lower bound for domination number in graphs, establishes NP-completeness of certain decision problems, and provides bounds and properties for specific graph classes, including outerplanar graphs and trees.

Contribution

It proves NP-completeness of equality decision for the Slater number, introduces efficient bounds for domination numbers in various graph classes, and generalizes the Slater number concept to total domination.

Findings

Deciding equality of domination number and Slater number is NP-complete.

Efficiently deciding whether (G)>(G) or (G)((G)) is polynomial.

Bounds on domination number for graphs in al(,) and outerplanar graphs.

Abstract

Let $G$ be a graph with degree sequence $d_1\geq \ldots \geq d_n$. Slater proposed $s\ell(G)=\min\{ s: (d_1+1)+\cdots+(d_s+1)\geq n\}$ as a lower bound on the domination number $\gamma(G)$ of $G$. We show that deciding the equality of $\gamma(G)$ and $s\ell(G)$ for a given graph $G$ is NP-complete but that one can decide efficiently whether $\gamma(G)>s\ell(G)$ or $\gamma(G)\leq \left(\left\lceil\ln \left(\frac{n(G)}{s\ell(G)}\right)\right\rceil+1\right)s\ell(G)$. For real numbers $\alpha$ and $\beta$ with $\alpha\geq \max\{ 0,\beta\}$, let ${\cal G}(\alpha,\beta)$ be the class of non-null graphs $G$ such that every non-null subgraph $H$ of $G$ has at most $\alpha n(H)-\beta$ many edges. Generalizing a result of Desormeaux, Haynes, and Henning, we show that $\gamma(G)\leq (2\alpha+1)s\ell(G)-2\beta$ for every graph $G$ in ${\cal G}(\alpha,\beta)$ with $\alpha \leq \frac{3}{2}$. Furthermore, we show that $\gamma(G)/s\ell(G)$ is bounded for graphs $G$ in ${\cal G}(\alpha,\beta)$ if and only if $\alpha<2$. For an outerplanar graph $G$ with $s\ell(G)\geq 2$, we show $\gamma(G)\leq 6s\ell(G)-6$. In analogy to $s\ell(G)$, we propose $s\ell_t(G)=\min\{ s: d_1+\cdots+d_s\geq n\}$ as a lower bound on the total domination number. Strengthening results due to Raczek as well as Chellali and Haynes, we show that $s\ell_t(T)\geq \frac{n+2-n_1}{2}$ for every tree $T$ of order $n$ at least $2$ with $n_1$ endvertices.