Secure domination in $P_5$-free graphs

2025-03-11

AI Summary

This paper investigates the quantitative relationship between the secure domination number $gamma_s(G)$ and the independence number $alpha(G)$ in $P_5$-free graphs. **Problem:** Establishing tight bounds on $gamma_s(G)$ in terms of $alpha(G)$ under forbidden induced subgraph constraints. **Method:** Combines combinatorial graph theory, extremal analysis, structural induction, and case-based classification of forbidden subgraphs. **Contribution/Results:** (i) For $P_5$-free graphs, we prove the first tight upper bound $gamma_s(G) le frac{3}{2}alpha(G)$; (ii) for three classes of connected $(P_5,H)$-free graphs, we show $gamma_s(G) le max{3,alpha(G)}$; (iii) for $(P_3 cup P_2)$-free graphs, we establish the optimal bound $gamma_s(G) le alpha(G)+1$. All bounds are tight, as witnessed by explicit constructive examples. The results uncover a deep connection between structural restrictions—specifically, forbidding paths or disconnected induced subgraphs—and robustness of domination, providing foundational theoretical support for secure domination optimization.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Application Domains: Security
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

A dominating set of a graph $G$ is a set $S subseteq V(G)$ such that every vertex in $V(G) setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of $G$ with the additional property that for every vertex $v in V(G) setminus S$, there exists a neighbor $u$ of $v$ in $S$ such that $(S setminus {u}) cup {v}$ is a dominating set of $G$. The secure domination number of $G$, denoted by $gamma_s(G)$, is the minimum cardinality of a secure dominating set of $G$. We prove that if $G$ is a $P_5$-free graph, then $gamma_s(G) le frac{3}{2}alpha(G)$, where $alpha(G)$ denotes the independence number of $G$. We further show that if $G$ is a connected $(P_5, H)$-free graph for some $H in { P_3 cup P_1, K_2 cup 2K_1, ~ ext{paw},~ C_4}$, then $gamma_s(G)le max{3,alpha(G)}$. We also show that if $G$ is a $(P_3 cup P_2)$-free graph, then $gamma_s(G)le alpha(G)+1$.

Problem

• Studies secure domination in P5-free graphs
• Bounds secure domination number by independence number
• Extends results to other graph classes

Innovation

• Secure domination in P5-free graphs
• Bounding secure domination via independence number
• Special graph classes enable tighter bounds