Domination and packing in graphs

TL;DR

Let $\gamma(G)$ and $\rho(G)$ denote the domination and packing numbers of a graph $G$. The paper investigates constant-factor bounds $\gamma(G) \le c\,\rho(G)$ across graph classes, contributing toward the Henning subcubic conjecture. It proves $\gamma(G) \le \frac{120}{49}\,\rho(G)$ for bicubic graphs, $\gamma(G) \le 3\rho(G)$ for maximal outerplanar graphs, and $\gamma(G) \le 2\rho(G)$ for biconvex graphs (the latter bound being tight). The methods combine known cubic graph bounds with class-specific structural decompositions and duality arguments, including a complete bipartite decomposition for biconvex graphs and properties of outerplanar duals. These results advance understanding of how domination can be efficiently bounded by packing in structured graph families and point toward broader constant-factor bounds in subcubic graphs.

Abstract

Given a graph~$G$, the domination number, denoted by~$γ(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose pairwise distance is at least three. The packing number~$ρ(G)$ of~$G$ is the maximum cardinality of one such set. Furthermore, the inequality~$ρ(G) \leq γ(G)$ is well-known. Henning et al.\ conjectured that~$γ(G) \leq 2ρ(G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~${γ(G) \leq \frac{120}{49}ρ(G)}$ if~$G$ is a bipartite cubic graph. We also show that $γ(G) \leq 3ρ(G)$ if~$G$ is a maximal outerplanar graph, and that~$γ(G) \leq 2ρ(G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight.

Figures (6)

• Figure 1: The sets $W$, $T$ and the graph $H$ in the proof of Lemma \ref{['lemma:packingcubicn7']}.
• Figure 2: A maximal outerplanar graph $G$ is represented by the white circles and by the dashed edges. The nodes of $G^{*}$ are the black circles. By selecting a node $r$, we obtain the directed tree $T$. The arcs with curvilinear shape are in $T_u$, and $G_u$ is the induced subgraph obtained from the union of the shaded triangles. In this case $h(u) = 4$.
• Figure 3: The structure of a biconvex graph $G$.
• Figure 4: The graph $G'_3$.
• Figure 5: The sun graph satisfies that $\gamma(G) = 2\rho(G)$.

Theorems & Definitions (27)

• Conjecture 1: Henning et al., 2011
• Lemma 2: KostochkaS09
• Lemma 3
• proof
• Claim 1
• proof : Proof of Claim
• Theorem 4: BondyM08
• Theorem 5
• Theorem 6
• Proposition 7: BrandstadtCD98