Best possible upper bounds on the restrained domination number of cubic graphs

Abstract

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G - S$ obtained by removing all vertices in $S$ is isolate-free. The domination number $γ(G)$ and the restrained domination number $γ_{r}(G)$ are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of $G$. Let $G$ be a cubic graph of order~$n$. A classical result of Reed [Combin. Probab. Comput. 5 (1996), 277--295] states that $γ(G) \le \frac{3}{8}n$, and this bound is best possible. To determine a best possible upper bound on the restrained domination number of $G$ is more challenging, and we prove that $γ_{r}(G) \le \frac{2}{5}n$.

Figures (16)

• Figure 1: The Petersen graph $G$
• Figure 2: The family ${\cal B}_{{\rm rdom}}$
• Figure 3: A subgraph in the proof of Claim \ref{['claim.12.1']}
• Figure 4: A subgraph in the proof of Claim \ref{['claim.12.1']}
• Figure 5: A subgraph in the proof of Claim \ref{['claim.12.1']}

Theorems & Definitions (75)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Claim 1
• Claim 2
• Claim 3
• Claim 4
• Claim 5