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Complexity of Eternal-Dominating-Set on perfect graphs

Determine the computational complexity of Eternal-Dominating-Set on perfect graphs, i.e., given a perfect graph G and a set of guards D ⊆ V(G), decide whether D is an eternal dominating set of G.

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Background

The paper contrasts two closely related problems: Eternal-Dominating-Number and Eternal-Dominating-Set. While Eternal-Dominating-Number is polynomial-time solvable on perfect graphs (via α(G)=γ∞(G)=θ(G)), Eternal-Dominating-Set has been shown EXP-complete in general graphs.

However, for perfect graphs specifically, the complexity of Eternal-Dominating-Set remains unresolved, leaving a gap between known results for the number problem and the set verification problem.

References

In particular, on perfect graphs, the complexity of Eternal-Dominating-Set is still open contrary to the complexity of Eternal-Dominating-Number.

— Fast winning strategies for the attacker in eternal domination (2401.10584 - Bagan et al., 19 Jan 2024) in Section 1 (Introduction)