A Note on the Complexity of Defensive Domination

TL;DR

The paper studies defending a graph against any $k$-vertex attack using an $\ell$-defense, formalized as a two-level quantified problem within the polynomial hierarchy. It proves $$Σ^P_{2}$$-completeness for DefensiveDominatingSet and, via a natural multiset extension, for DefensiveDominatingMultiset, using a reduction from CliqueNodeDeletion. Conversely, it shows that DefensiveDominatingMultiset is solvable in polynomial time on interval graphs through a greedy interval-based algorithm, revealing a sharp contrast between general graphs and interval-graph instances. The work also discusses implications for related notions such as GoodDefense/BadDefense, and outlines open questions, notably the complexity of DefensiveDominatingSet on interval graphs and potential tractability in planar graphs or under treewidth-based approaches.

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 $Σ^P_2$. 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 $Σ^P_2$-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 $Σ^P_2$-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.

Figures (1)

• Figure 2: Rutenburg's reduction. The part of $G$ corresponding to the clauses $C_1 \equiv (x_1 \vee \neg x_2 \vee y_1)$ and $C_2 \equiv (\neg x_2 \vee \neg x_3 \vee \neg y_3)$. Ugly vertices are represented by crossed circles.

Theorems & Definitions (9)

• Lemma 2: Theorem 2.3 in Ekim, Farley, Proskurowski EkimFP20
• Theorem 3
• Theorem 3
• Theorem 4
• Theorem 5: Theorem 6 in Rutenburg Rutenburg86
• Theorem 5
• Lemma 8
• Theorem 8
• Theorem 8: Theorem 6 in Rutenburg Rutenburg86