A Note on the Complexity of Defensive Domination

2025-04-19

AI Summary

This paper investigates the Defensive Domination problem on graphs: given a graph $G$ and positive integers $k,l$, determine whether there exists a vertex set $D$ of size at most $l$ such that, for every attack set $A$ of size at most $k$, $D$ can defend by covering $A$ via its closed neighborhood. We establish, for the first time, that this problem is $Sigma_2^P$-complete. Subsequently, we introduce a variant allowing multiple selections and devise the first polynomial-time algorithm for it on interval graphs. Our main contributions are threefold: (i) pinpointing the exact computational complexity class of the original problem; (ii) extending the boundary of graph classes for which defensive domination is polynomially solvable; and (iii) providing a foundational complexity benchmark for robust network design, disaster response planning, and redundant system deployment.

Technology Category

• Application Domains: Security
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In a graph G, a k-attack A is any set of at most k vertices and l-defense D is a set of at most l vertices. We say that defense D counters attack A if each a in A can be matched to a distinct defender d in D with a equal to d or a adjacent to d in G. In the defensive domination problem, we are interested in deciding, for a graph G and positive integers k and l given on input, if there exists an l-defense that counters every possible k-attack on G. Defensive domination is a natural resource allocation problem and can be used to model network robustness and security, disaster response strategies, and redundancy designs. The defensive domination problem is naturally in the complexity class $Sigma^P_2$. The problem was known to be NP-hard in general, and polynomial-time algorithms were found for some restricted graph classes. In this note we prove that the defensive domination problem is $Sigma^P_2$-complete. We also introduce a natural variant of the defensive domination problem in which the defense is allowed to be a multiset of vertices. This variant is also $Sigma^P_2$-complete, but we show that it admits a polynomial-time algorithm in the class of interval graphs. A similar result was known for the original setting in the class of proper interval graphs.

Problem

• Determining if an l-defense counters all k-attacks in graphs
• Proving defensive domination is Σ^P_2-complete complexity
• Solving multiset defense variant in interval graphs efficiently

Innovation

• Proves defensive domination is Σ^P_2-complete
• Introduces multiset defense variant
• Provides polynomial-time algorithm for interval graphs