$2$-limited broadcast domination in cubic graphs

TL;DR

The paper proves that for cubic graphs, the $2$-limited broadcast domination number satisfies $\gamma_{b,2}(G)\le \frac{|V(G)|}{3}$, confirming a conjecture of Henning, MacGillivray, and Yang. Central to the proof is a weight framework: assign vertex weights by degree and add a penalty for bad components, yielding the global bound $9\gamma_{b,2}(G)\le \omega(G)$ for subcubic graphs. Through a minimal counterexample argument and a sequence of structural lemmas, the authors show that a hypothetical counterexample must have highly constrained local structures around $3$-vertices and $4$-cycles, which ultimately leads to a contradiction. Consequently, the bound holds for cubic graphs, and the approach links to broader questions related to domination in cubic and near-cubic graphs.

Abstract

For a graph $G$, a function $f:V(G) \to \{0,1,2\}$ is called a $2$-limited dominating broadcast on $G$ if for every vertex $u$, there exists a vertex $v$ such that $f(v)>0$ and the distance between $u$ and $v$ in $G$ is at most $f(v)$. The {\it cost} of $f$ means the value $\sum_{v\in V(G)}f(v)$, and the {\it $2$-limited broadcast domination number} of $G$, denoted by $γ_{b,2}(G)$, is the cost of a $2$-limited dominating broadcast on $G$ with minimum cost. Henning, MacGillivray, and Yang (2020) conjectured that $γ_{b,2}(G)\leq \frac{|V(G)|}{3}$ for every cubic graph $G$. In this paper, we confirm the conjecture.

Figures (16)

• Figure 1: The graph $K^*_4$
• Figure 2: The subgraph mentioned in Lemma \ref{['Lem:b(G)=0']}
• Figure 3: The choices of the vertex $v^*$
• Figure 4: Subgraphs mentioned in Lemma \ref{['Lem:partial=2']}
• Figure 5: Subgraphs mentioned in Lemma \ref{['Lem:no_pendant']}

Theorems & Definitions (69)

• Conjecture 1.1: HMY2020
• Theorem 1.2: HMY2020
• Theorem 1.3: park20232
• Theorem 1.4
• Theorem 1.5
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof