Cop-win graph¶
A graph on which one pursuer has a strategy that guarantees capture of one evader in the alternating vertex-movement game.
Core Idea¶
For finite undirected reflexive graphs the class is characterized by dismantlability: repeatedly remove a vertex whose closed neighborhood is contained in another's until one vertex remains. The cop chooses first, the robber responds and both move along an edge or remain; domination supplies a retraction that converts a winning strategy on the smaller graph into one on the original. 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¶
Cop-win graph belongs to graph pursuit games and is useful where the analyst can specify the typed graph pursuit games carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the graph class and loop convention, player counts and initial-choice order, legal move and information rules, capture condition, finite assumption, winning strategy and dismantling or constructibility certificate are explicit. The scope is broad within that domain but bounded by the need for the graph class and loop convention, player counts and initial-choice order, legal move and information rules, capture condition, finite assumption, winning strategy and dismantling or constructibility certificate are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the graph class and loop convention, player counts and initial-choice order, legal move and information rules, capture condition, finite assumption, winning strategy and dismantling or constructibility certificate 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.
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 Cop-win graph. Cop-win graph 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 pursuit games carrier, including its objects, 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 the graph class and loop convention, player counts and initial-choice order, legal move and information rules, capture condition, finite assumption, winning strategy and dismantling or constructibility certificate are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of graph pursuit games because they reuse the typed graph pursuit games carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, The cop chooses first, the robber responds and both move along an edge or remain; domination supplies a retraction that converts a winning strategy on the smaller graph into one on the original., and type the carrier, state every parameter and convention in the definition, test that the graph class and loop convention, player counts and initial-choice order, legal move and information rules, capture condition, finite assumption, winning strategy and dismantling or constructibility certificate are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Cop-win graph Domain-specific
Parents (1) — more general patterns this builds on
-
Cop-win graph is a kind of Coverage / Reachability Prime
The proposed strict upward parent is
prime:coverage_reachability.
Hierarchy paths (2) — routes to 2 parentless roots
- Cop-win graph → Coverage / Reachability → Completeness
- Cop-win graph → Coverage / Reachability → Surjectivity → Function (Mapping)
Neighborhood in Abstraction Space¶
Cop-win graph sits in a crowded region of the domain-specific corpus (37th 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
- Cop number — 0.95
- Entanglement (graph measure) — 0.91
- Eternal dominating set — 0.91
- Move by nature — 0.89
- Matching (graph theory) — 0.89
Computed from structural-signature embeddings · 2026-09-08