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Secure Total Domination Number in Maximal Outerplanar Graphs

Yasufumi Aita, Toru Araki

TL;DR

This work determines the secure total domination number $\gamma_{st}$ for maximal outerplanar graphs by combining structural analysis with inductive constructions. It establishes sharp bounds $\left\lceil (n+2)/3 \right\rceil \le \gamma_{st}(G) \le \left\lfloor 2n/3 \right\rfloor$ for graphs of order $n$, and provides explicit extremal families to demonstrate tightness. The upper bound is achieved by the construction family $H_k$ where $\gamma_{st}(H_k)=\left\lfloor 2n/3 \right\rfloor$, while the lower bound is tight for the family $G_k$ with $\gamma_{st}(G_k)=\left\lceil (n+2)/3 \right\rceil$. These results give a complete, constructive characterization of STDS in maximal outerplanar graphs and highlight how the outerplanar structure governs secure total domination parameters.

Abstract

A subset $S$ of vertices in a graph $G$ is a secure total dominating set of $G$ if $S$ is a total 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 total dominating set of $G$. We show that if $G$ is a maximal outerplanar graph of order $n$, then $G$ has a total secure dominating set of size at most $\lfloor 2n/3 \rfloor$. Moreover, if an outerplanar graph $G$ of order $n$, then each secure total dominating set has at least $\lceil (n+2)/3 \rceil$ vertices. We show that these bounds are best possible.

Secure Total Domination Number in Maximal Outerplanar Graphs

TL;DR

This work determines the secure total domination number for maximal outerplanar graphs by combining structural analysis with inductive constructions. It establishes sharp bounds for graphs of order , and provides explicit extremal families to demonstrate tightness. The upper bound is achieved by the construction family where , while the lower bound is tight for the family with . These results give a complete, constructive characterization of STDS in maximal outerplanar graphs and highlight how the outerplanar structure governs secure total domination parameters.

Abstract

A subset of vertices in a graph is a secure total dominating set of if is a total dominating set of and, for each vertex , there is a vertex such that is an edge and is also a total dominating set of . We show that if is a maximal outerplanar graph of order , then has a total secure dominating set of size at most . Moreover, if an outerplanar graph of order , then each secure total dominating set has at least vertices. We show that these bounds are best possible.
Paper Structure (4 sections, 7 theorems, 6 equations, 4 figures)

This paper contains 4 sections, 7 theorems, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Structure of maximal outerplanar graphs
  3. Upper bound
  4. Lower bound

Key Result

Proposition 2.1

Every maximal outerplanar graph contains at least two vertices of degree 2.

Figures (4)

  • Figure 1: Eight induced subgraphs in a maximal outerplanar graph.
  • Figure 2: Minimum STDSs for maximal outerplanar graphs of $n \leq 6$ vertices. The gray vertices are members of STDSs.
  • Figure 3: A maximal outerplanar graph $H_{k}$ when $k=4$. The set of the gray vertices form a minimum STDS of $H_{k}$.
  • Figure 4: A outerplanar graph $G_{k}$ when $k=4$. The set of the gray vertices form an STDS of $G_{4}$.

Theorems & Definitions (10)

  • Proposition 2.1: li16
  • Proposition 2.2: orourke87
  • Theorem 2.3
  • Theorem 3.1
  • Proposition 3.2: klostermeyer08:_secur
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof : Proof of Claim 1.
  • proof : Proof of Claim 2.