Fast winning strategies for the attacker in eternal domination

TL;DR

The paper examines the Fast-Strategy problem in eternal domination games, formalizing $t_G(D)$ as the earliest turn when Attacker can force a Defender loss and analyzing the computational complexity across graph classes. It proves PSPACE-hardness on several restricted classes while delivering polynomial-time algorithms for trees and cographs, and it links the tree case to contracted treedepth via arenas. A cotree-based approach and reservist variant underpin the cograph result, and the work includes a thorough parameterized complexity analysis with respect to the number of guards $g$ and turns $t$, including kernelization, XP/FPT results, and first-order definability. Overall, the results illuminate a nuanced complexity landscape for attacker strategies in dynamic graph surveillance and provide practical algorithms for key graph families.

Abstract

Dominating sets in graphs are often used to model some monitoring of the graph: guards are posted on the vertices of the dominating set, and they can thus react to attacks occurring on the unguarded vertices by moving there (yielding a new set of guards, which may not be dominating anymore). A dominating set is eternal if it can endlessly resist to attacks. From the attacker's perspective, if we are given a non-eternal dominating set, the question is to determine how fast can we provoke an attack that cannot be handled by a neighboring guard. We investigate this question from a computational complexity point of view, by showing that this question is PSPACE-hard, even for graph classes where finding a minimum eternal dominating set is in P. We then complement this result by giving polynomial time algorithms for cographs and trees, and showing a connection with tree-depth for the latter. We also investigate the problem from a parameterized complexity perspective, mainly considering two parameters: the number of guards and the number of steps.

Figures (5)

• Figure 1: Above, an example of a winning strategy for Attacker in 3 turns, depending on the answer of Defender. Each number represents a guard, and the attack is circled. Below, an eternal dominating set with 3 guards. Each guard protects its own clique.
• Figure 2: On the left, the graph $P_7$ of treedepth 3. On the right, an optimal td-decomposition of $P_7$
• Figure 3: A partial representation of the reduction from $(G, k)$ where $G$ is a graph with 4 vertices $\{v_1, v_2, v_3, v_4\}$ and two edges $v_1v_2$ and $v_2v_3$ and $k = 3$. Every vertex in $S_{2,3}$ is connected to $v_2$ and $v_3$. The vertices above the dashed line are guarded. The length of the game is $t = 2k+1=7$.
• Figure 4: A partial representation of the reduction from the formula $(x_1 \vee \overline{y_1}) \wedge (\overline{x_1} \vee x_2 \vee y_2)$. Only two checkers are represented, one for the variable $y_2$ and the other for the clause $C_2 = \overline{x_1} \vee x_2 \vee y_2$. The vertices above the dashed line are guarded and forms a clique. All vertices of $V_{y_2}$ are connected to the 8 vertices in the thick rectangle. All vertices of $F_2$ are connected to the vertices in the hatched zones. The length of the game is $t = k+M = 34$.
• Figure 5: On the left, a tree with an arena of contracted treedepth 3. Each grayed vertex is guarded, and the attack is circled. On the right, the same tree after the Defender has moved the guard from left to the attack. A smaller arena of contracted treedepth 2 is created (its contracted tree is a path on three vertices).

Theorems & Definitions (65)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['eds-bipartite']}
• Lemma 5
• proof
• Theorem 6