An improved upper bound for the domination number of a graph

TL;DR

The paper investigates improving upper bounds on the domination number $\gamma(G)$ using the multiplicative Nordhaus–Gaddum bound $\gamma(G)\gamma(\overline{G}) \le n$, establishing that for any graph of order $n$ at least one of $\gamma(G)$ or $\gamma(\overline{G})$ is at most $\left\lfloor\sqrt{n}\right\rfloor$. It provides a sharp necessary condition $\Delta \ge \lceil\sqrt{n}\rceil-1$ for graphs with $\gamma(G) \le \left\lfloor\sqrt{n}\right\rfloor$ and a complete structural characterization of connected graphs achieving $\gamma(G) = \left\lfloor\sqrt{n}\right\rfloor$ via a special graph family $\mathscr{F}$ built from $H_1,H_2$ with a partition and a consistency Condition $C$. The work also gives several sufficient conditions ensuring $\gamma(G) \le \left\lfloor\sqrt{n}\right\rfloor$, introduces domination-type classifications and the $SCC(2)$ class, and provides a polynomial-time method to decide which domination type applies when $G \notin SCC(2)$, with a conjecture extending to $G \in SCC(2)$. The conclusions highlight practical implications for network design and propose directions for identifying other parameters amenable to Nordhaus–Gaddum-type improvements and for studying the structure of graphs meeting the bound.

Abstract

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $γ(G) \le \lfloor\sqrt{n}\rfloor$ which gives a substantially improved upper bound. In this paper, we give a condition necessary for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$, and some conditions sufficient for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$. We also present a characterization of all connected graphs $G$ of order $n$ with $γ(G) = \lfloor\sqrt{n}\rfloor$. Further, we prove that for a graph $G$ not satisfying $rad(G)=diam(G)=rad(\overline{G})=diam(\overline{G})=2$, deciding whether $γ(G) \le \lfloor\sqrt{n}\rfloor$ or $γ(\overline{G}) \le \lfloor\sqrt{n}\rfloor$ can be done in polynomial time. We conjecture that this decision problem can be solved in polynomial time for any graph $G$.

Figures (1)

• Figure 1: An example of a graph in the family $\mathscr{F}$

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• proof
• Definition 2.2
• Theorem 2.3