$m$-Eternal Dominating Set Problem on Subclasses of Chordal Graphs

A dominating set of a graph G(V, E) is a set of vertices D\subseteq V such that every vertex in V\D has a neighbor in D. An eternal dominating set extends this concept by placing mobile guards on the vertices of D. In response to an infinite sequence of attacks on unoccupied vertices, a guard can move to the attacked vertex from an adjacent position, ensuring that the new guards configuration remains a dominating set. In the one (all) guard(s) move model, only one (multiple) guard(s) moves(may move) per attack. The set of vertices representing the initial configuration of guards in one(all) guard move model is the eternal dominating set (m-eternal dominating set) of G. The minimum size of such a set in one(all) guard move model is called the eternal domination number (m-eternal domination number) of G, respectively. Given a graph G and an integer k, the m-Eternal Dominating Set asks whether G has an m-eternal dominating set of size at most k. In this work, we focus mainly on the computational complexity of m-Eternal Dominating Set in subclasses of chordal graphs. For split graphs, we show a dichotomy result by first designing a polynomial-time algorithm for K1,t-free split graphs with t\le 4, and then proving that the problem becomes NP-complete for t\ge 5. We showed that the problem is NP-hard on undirected path graphs. Moreover, we exhibit the computational complexity difference between the variants by showing the existence of two graph classes such that, in one, both Dominating Set and m-Eternal Dominating Set are solvable in polynomial time while Eternal Dominating Set is NP-hard, whereas in the other, Eternal Dominating Set is solvable in polynomial time and both Dominating Set and m-Eternal Dominating Set are NP-hard. Finally, we present a graph class where Dominating Set is NP-hard, but m-Eternal Dominating Set is efficiently solvable.

💡 Research Summary

This paper investigates the computational complexity of the m‑eternal dominating set problem, a dynamic variant of the classic dominating set problem, within two important subclasses of chordal graphs: split graphs and undirected path graphs. A dominating set D of a graph G ensures that every vertex outside D has a neighbor in D. In the eternal domination framework, guards are placed on the vertices of D and must respond to an infinite sequence of attacks on unoccupied vertices, moving along edges while always maintaining a dominating configuration. The “one‑guard‑move” model permits only a single guard to move per attack, whereas the “all‑guards‑move” (m‑model) allows every guard to relocate simultaneously, still respecting the rule that at most one guard occupies any vertex. The minimum size of a guard set that can sustain such a defense is the m‑eternal domination number γ∞^m(G).

The authors first focus on split graphs, which can be partitioned into a clique C and an independent set I. They introduce the parameter Δ_I(G), the maximum number of neighbors a vertex in C has within I, and study K₁,t‑free split graphs (graphs that do not contain a star K₁,t as an induced subgraph). For t ≤ 4 they design a deterministic algorithm that computes γ∞^m(G) in O(|V|^{3/2}) time, improving on the previously known O(|V|^{5/2}) bound. The algorithm exploits the limited interaction between C and I imposed by the K₁,t‑free condition, reduces the problem to a matching computation, and systematically tests feasible guard placements.

In contrast, when t ≥ 5 the problem becomes NP‑complete. The hardness proof reduces from Exact‑3‑Cover, constructing a split graph where each element and each 3‑set correspond to vertices in C and I, respectively, and the presence of a K₁,5 forces any feasible guard configuration to encode a perfect cover. This establishes that even the more permissive m‑model remains computationally intractable for sufficiently large stars.

Next, the paper turns to undirected path graphs, which are intersection graphs of paths in a tree and sit between interval graphs (where the problem is polynomial) and general chordal graphs (where it is NP‑hard). By a reduction from 3‑Dimensional Matching, the authors show that determining γ∞^m(G) for undirected path graphs is NP‑hard. The reduction maps each triple of the matching instance to a specific arrangement of paths, ensuring that a successful guard strategy corresponds exactly to a valid 3‑dimensional matching. This result closes the complexity gap for this intermediate class.

Beyond these core results, the authors explore the relationship among three related problems: Dominating Set (DS), Eternal Dominating Set (EDS), and m‑Eternal Dominating Set (m‑EDS). They exhibit two graph families that illustrate contrasting complexity landscapes. In the first family, both DS and m‑EDS are solvable in polynomial time, while EDS remains NP‑hard, demonstrating that allowing all guards to move can dramatically simplify the problem. In the second family, EDS is polynomial, but both DS and m‑EDS are NP‑hard, showing that the one‑guard‑move restriction can sometimes make the problem easier than the static version.

Finally, the paper presents a construction where DS is NP‑hard but m‑EDS is polynomial. Starting from any graph G, they attach n disjoint copies of a five‑vertex path P₅, connecting the middle vertex of each copy to a distinct vertex of G. In this augmented graph, any optimal guard placement for the m‑model places a guard on each middle vertex of the attached P₅’s, guaranteeing immediate defense of any attack and yielding a trivial computation of γ∞^m(G), while the underlying DS problem retains its hardness.

Overall, the work maps out the complexity terrain of m‑eternal domination across several chordal subclasses, provides tight algorithmic and hardness bounds, and highlights how subtle changes in guard movement rules or graph structure can shift a problem from tractable to intractable. The results have implications for network security, resource allocation, and dynamic monitoring scenarios where mobile defenders must continuously protect a system against adversarial actions.