A Lower bound for Secure Domination Number of an Outerplanar Graph

TL;DR

This work analyzes the secure domination number $\gamma_{s}(G)$ for outerplanar graphs and proves a universal lower bound $\gamma_{s}(G) \ge (n+4)/5$ for graphs with $n\ge 4$, with the bound shown to be tight. The approach combines a careful partition of a minimum secure dominating set by external private neighbors, a counting argument, and a bipartite-subgraph bound, to force $y \le 4x-4$ where $x=|S|$ and $y=|V(G)\setminus S|$. The tightness is established via an explicit extremal construction $G_k$ on $n=5k+1$ vertices achieving $\gamma_{s}(G_k)=(n+4)/5$, and a spanning-subgraph characterization showing that equality necessitates containing $G_k$. These results provide a sharp baseline for secure domination in outerplanar graphs and a blueprint for identifying extremal structures.

Abstract

A subset $S$ of vertices in a graph $G$ is a secure dominating set of $G$ if $S$ is a dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup \{u\}$ is also a dominating set of $G$. The secure domination number of $G$, denoted by $γ_{s}(G)$, is the cardinality of a smallest secure dominating sets of $G$. In this paper, we prove that for any outerplanar graph with $n \geq 4$ vertices, $γ_{s}(G) \geq (n+4)/5$ and the bound is tight.

Figures (1)

• Figure 1: The outerplanar graph $G_{k}$ when $k=4$.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 2.1: cockayne08:_protec
• Theorem 2.2
• Lemma 2.3
• Theorem 3.1
• proof