Some Comments on the Slater number

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.