Secure Total Domination Number in Maximal Outerplanar Graphs

Core Concepts

The author establishes upper and lower bounds for the secure total domination number in maximal outerplanar graphs, proving that these bounds are sharp.

Abstract

The content discusses the concept of secure total domination sets in graphs, focusing on maximal outerplanar graphs. It presents upper and lower bounds for the secure total domination number, demonstrating their optimality through detailed proofs and examples. The study delves into various aspects of secure domination, providing insights into the complexity and properties of these dominating sets in graph theory.

Stats

G has a total secure dominating set of size at most ⌊2n/3⌋. Each secure total dominating set in an outerplanar graph has at least ⌈(n+2)/3⌉ vertices. For any maximal outerplanar graph G of n ≥ 3 vertices, γst(G) ≤ ⌊2n/3⌋. For an outerplanar graph G of n ≥ 3 vertices, γst(G) ≥ ⌈(n + 2)/3⌉. The graph Hk is order 3k and has k vertices of degree 2.

Quotes

"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 ⌊2n/3⌋." "Let S be a total dominating set of G. For u /∈ S and v ∈ S, if uv ∈ E(G) and (S \ {v}) ∪ {u} is also a total dominating set of G..." "In this paper, we give sharp upper and lower bounds for the secure total domination number of maximal outerplanar graphs."

Key Insights Distilled From

Secure Total Domination Number in Maximal Outerplanar Graphs

by Yasufumi Ait... at arxiv.org 03-07-2024

https://arxiv.org/pdf/2403.03404.pdf

Deeper Inquiries

How do the results on secure total domination in maximal outerplanar graphs compare to other types of graphs

The results on secure total domination in maximal outerplanar graphs provide insights into the specific behavior of this graph class compared to others. In particular, the upper and lower bounds established for the secure total domination number shed light on the unique characteristics of maximal outerplanar graphs. These bounds demonstrate that for a maximal outerplanar graph of order n, the size of a total secure dominating set is at most ⌊2n/3⌋, while each secure total dominating set must have at least ⌈(n+2)/3⌉ vertices. In comparison to other types of graphs, such as trees or cographs where similar studies have been conducted, the findings highlight the distinct properties and complexities associated with maximal outerplanar graphs. The NP-hard nature of the secure total domination problem even within restricted graph classes like chordal bipartite and split graphs underscores the challenges posed by different graph structures.

What implications could the findings on secure domination have on real-world network security systems

The implications of these findings on secure domination extend beyond theoretical graph theory applications to real-world network security systems. Understanding how to efficiently identify and protect critical nodes within a network is crucial for enhancing overall system resilience against potential attacks or disruptions. By applying concepts from secure total domination in practical scenarios, network administrators can optimize resource allocation for monitoring and securing key components within complex networks. The ability to determine minimum sets that ensure complete coverage or protection can lead to more effective strategies for safeguarding sensitive information or maintaining operational continuity in various network environments. Furthermore, insights gained from studying secure domination in different graph models can inform cybersecurity protocols and defense mechanisms aimed at fortifying communication networks, financial systems, or critical infrastructure against cyber threats.

How might the concept of secure total domination apply to social networks or biological systems

The concept of secure total domination can find relevance in diverse fields beyond traditional graph theory applications, including social networks and biological systems. In social networks, identifying individuals who play pivotal roles in disseminating information or influencing group dynamics is essential for targeted interventions or marketing strategies. By leveraging ideas from secure dominance theory within social networks, researchers could pinpoint influential users whose engagement significantly impacts overall network behavior. This knowledge could be utilized to enhance advertising campaigns, viral content dissemination strategies, or community engagement initiatives by focusing efforts on key individuals identified through their role as part of a secured dominating set. Similarly, in biological systems such as gene regulatory networks or ecological interactions among species populations, understanding which elements exert control over system dynamics is vital for predicting responses to perturbations or disturbances. Applying principles of secured dominance could aid in identifying critical nodes responsible for maintaining stability within these complex systems and guide conservation efforts or therapeutic interventions targeting key regulatory elements.