Eternal Dominating Family in Graphs

• Eternal Dominating Family is a collection of configurations in graphs that ensures continuous domination under adversarial and iterative dynamics.
• The concept employs modular patterns and inequality chains to structure dynamic guard movements across various graph classes and grid networks.
• Algorithmic studies show NP-hardness in general cases with efficient linear-time solutions for specific graphs, highlighting applications in network surveillance and game-theoretic protection.

The concept of an Eternal Dominating Family encompasses families of configurations, sets, or strategies in graphs and related structures that maintain robust domination properties under adversarial, iterative, or infinite-duration dynamics. The formalism arises from graph theory, combinatorics, and topology, and serves both in discrete (finite or infinite graphs and grids) and abstract (binary relations, homomorphism inequalities) contexts. This article surveys its definition, structural paradigms, algorithmic and complexity aspects, and its key role in modern research.

1. Formal Definitions and Dynamic Domination

An Eternal Dominating Family is a set (or family) of configurations (often dominating sets in graphs) preserved under sequences of adversarial events such as attacks or selections. In the context of graphs, a dominating set $D \subseteq V(G)$ ensures that every vertex not in $D$ has a neighbor in $D$, and the eternal version requires a dynamic defense protocol:

• Eternal Domination Game: At each turn, an attacker chooses an unguarded vertex; the defender must move guards so the new configuration is still dominating and the attacked vertex is occupied. Variants such as m-eternal domination permit multiple guards to move simultaneously (Calamoneri et al., 12 Jul 2025).
• Eternal Dominating Family: The set of all possible configurations reachable by guard moves under the above dynamics, retaining domination after any infinite sequence of future attacks.

Extension to binary relations provides a framework for selection principles in topology: a dominating family in a relation $P = (A,B,R)$ is $Z \subseteq B$ such that $\forall a \in A$, $\exists b \in Z$ with $a R b$; generalizations consider products and persistent domination under all subfamilies (Dias et al., 2014).

2. Structural Results and Bounds in Graph Classes

Eternal domination exhibits distinctive behaviors across graph families:

• Inequality Chains: Key domination parameters satisfy $$\gamma(G) \leq \alpha(G) \leq \gamma^{(\infty)}(G) \leq \theta(G)$$, with $\gamma$ as domination number, $\alpha$ independence number, $\theta$ clique covering number (Klostermeyer et al., 2014). For m-eternal domination, $$\gamma(G) \leq \gamma_m^{(\infty)}(G) \leq \alpha(G)$$.
• Characterizations: In bipartite/trianlge-free graphs, only specific structures allow the minimal possible eternal domination numbers (e.g., only certain matchings removed from complete bipartite graphs yield $\gamma_m^{(\infty)}=2$) (Klostermeyer et al., 2014). For trees, reduction operations (deleting stems and leaves) yield that $\gamma_m^{(\infty)}(T) = \theta(T)$ if and only if $T$ can be reduced to a simple star.
• Interval Graphs: For games where all guards may move, the eternal domination and clique-connected cover numbers coincide, and a linear algorithm computes optimal eternal dominating sets (Rinemberg et al., 2018).
• Grids and Products: On large rectangular grids, the m-eternal and static domination numbers are nearly equal; for $P_m \Box P_n$, both are asymptotically $\frac{mn}{5}$ for large $m,n$ (Lamprou et al., 2016). In strong grids ($P_n \boxtimes P_m$), the eternal domination number is $\frac{mn}{7}+O(m+n)$, improving previous bounds for slender grids (Gagnon et al., 2020).

3. Algorithmic and Complexity Perspectives

Determining eternal domination numbers and families is computationally challenging:

• NP-Hardness: The m-eternal domination problem is NP-hard, including for bipartite graphs of diameter four. Roman and Italian eternal domination variants inherit this hardness (Calamoneri et al., 12 Jul 2025).
• Tree-Specific Algorithms: Efficient procedures exist for trees: for less than the required number of guards, an attack sequence guaranteeing the attacker's win is computable in at most $n$ moves, exploiting "neo-colonization" partitions and deficit analysis (Blažej et al., 2022).
• Parameterized Complexity: Fast winning strategies for attackers (minimizing turns until victory) are PSPACE-hard on general graphs, polynomial-time for trees and cographs via decomposition (arena, cotree), and fixed-parameter tractable in specific cases (Bagan et al., 19 Jan 2024).
• Linear-Time for Interval Graphs: The eternal dominating set can be computed via interval endpoint sweeps, offering practical applicability where interval models are given (Rinemberg et al., 2018).

4. Dynamic Domination, Modular and Rotation Strategies

Families of robust dominating sets are constructed via combinatorial or geometric patterns:

• Modular Patterns: On the infinite square grid, using $f(x,y)=x+2y\mod5$ partitions the grid into "perfect" dominating sets that can be rotated or translated upon each attack to perpetuate domination (Lamprou et al., 2016).
• Alternating Strategies: On strong grids, alternating between two families of periodic dominating sets (e.g., $D$ and $D'$ defined via modular or recursive rules) allows local or modular defense (partitioning into subgrids and planning movement within regions), which is both efficient and scalable (Gagnon et al., 2020).
• Translation and Partitioning: For infinite regular grids (square, octagonal, hexagonal, triangular), recursive and congruence-based constructions yield strongly optimal dominating sets where each vertex's closed neighborhood is uniquely covered, and global translation after an attack restores the dominating configuration (Calamoneri et al., 12 Jul 2025).
• Reflection and Percolation: In homomorphism density contexts, layered percolating sequences through half-folding maps and cut involutions build dominating graphs that eternally dominate all their subgraphs (even-length paths, weakly norming graphs, reflection group constructions) (Conlon et al., 2023).

5. Variants and Extensions

Eternal domination admits generalizations and related families:

• Eternal Distance-$k$ Domination: Guards cover vertices within distance $k$ and can move up to $k$ per attack. The eternal distance-$k$ domination number for a path is $\lceil n/(k+1)\rceil$; for cycles, $\lceil n/(2k+1)\rceil$ (Cox et al., 2021).
• Fractional Eternal Domination: A relaxation allowing partial guard assignments, analyzed using flows and linear programming. It relates to classical domination via fractional parameters and is computable or bounded on classes like trees, split, strongly chordal, Kneser, abelian Cayley graphs, and products (Devvrit et al., 2023).

6. Implications and Applications

The concept of Eternal Dominating Families has consequences for:

• Fault-Tolerant Network Surveillance: Dynamic restoration of monitoring coverage after failures or adversarial events in sensor, communication, or defense networks.
• Distributed Resource Allocation: Ensuring sustainable coverage or connectivity under unpredictable conditions via reconfigurable resources.
• Game-Theoretic Network Protection: Model pursuit-evasion, recurrent defense, and parameterized analysis of adversarial tactics; for example, minimizing the number of turns in which an attacker can break domination (Bagan et al., 19 Jan 2024).
• Productivity and Preservation Theorems in Topology: Robustness of selection principles, duality in games, and preservation under products or subcollection operations (Dias et al., 2014).

7. Open Questions and Further Research

Current investigations target:

• Characterizing the gap between classical and eternal domination (e.g., is the difference always bounded by $O(n)$ on certain graph classes?).
• Tight bounds and structural analysis on complex infinite families, such as prisms, products, and generalized grid structures (Krim-Yee et al., 2019).
• Complexity boundaries—establishing tractability and hardness landscapes for various versions and parameters, including oriented domination on digraphs (Bagan et al., 2018), eternal Roman/Italian domination (Calamoneri et al., 12 Jul 2025), and first-order definable classes (Bagan et al., 19 Jan 2024).
• Extending methodologies (neo-colonization, modular partitioning, layered percolation) to broader combinatorial and topological domains.

Eternal Dominating Families thus unify multiple strands in combinatorics, topology, computational complexity, and practical network analysis by formalizing and analyzing robust, dynamic, and modular coverage strategies in both abstract and applied settings.