Eternal dominating set¶

A set of graph vertices occupied by mobile guards that can respond to every infinite sequence of vertex attacks while remaining a dominating set after each move.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4427
Origin domain: graph theoretic security games
Subdomain: graph theoretic security games

Core Idea¶

Eternal domination minimizes guards under a declared movement and attack model, with one-guard, all-guards, multiple-occupancy, and eviction variants producing different invariants. An adversary selects an attacked vertex, legal guards move along edges, one occupies the target when required, and the post-move configuration must dominate every vertex for the next attack. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Eternal dominating set belongs to graph theoretic security games and is useful where the analyst can specify the typed graph theoretic security games carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate graph, initial guard set, attack visibility and legality, movement model, occupancy, response rule, domination after every turn, adversary power, and minimum objective are explicit. The scope is broad within that domain but bounded by the need for graph, initial guard set, attack visibility and legality, movement model, occupancy, response rule, domination after every turn, adversary power, and minimum objective are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making graph, initial guard set, attack visibility and legality, movement model, occupancy, response rule, domination after every turn, adversary power, and minimum objective are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Eternal dominating set can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Eternal dominating set. Eternal dominating set compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed graph theoretic security games carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express graph, initial guard set, attack visibility and legality, movement model, occupancy, response rule, domination after every turn, adversary power, and minimum objective are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of graph theoretic security games because they reuse the typed graph theoretic security games carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, An adversary selects an attacked vertex, legal guards move along edges, one occupies the target when required, and the post-move configuration must dominate every vertex for the next attack., and type the carrier, state every parameter and convention in the definition, test that graph, initial guard set, attack visibility and legality, movement model, occupancy, response rule, domination after every turn, adversary power, and minimum objective are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Eternal dominating set Domain-specific

Parents (1) — more general patterns this builds on

Eternal dominating set is a kind of Coverage / Reachability Prime

The proposed strict upward parent is prime:coverage_reachability.

Hierarchy paths (2) — routes to 2 parentless roots

Eternal dominating set → Coverage / Reachability → Completeness

Show alternative path (1)

Neighborhood in Abstraction Space¶

Eternal dominating set sits in a crowded region of the domain-specific corpus (26^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Invariants & Constructions (49 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08