Cop number¶
The minimum number of pursuers required to guarantee capture of an evader in the standard cops-and-robber game on a graph.
Core Idea¶
The value depends on turn order, visibility, movement, capture and graph conventions; directed, fast-robber and invisible variants define different parameters. Cops choose initial vertices, the robber responds, players alternate along edges or wait and the least team size having a winning strategy over every robber response is assigned to the graph. 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 number belongs to graph theory and is useful where the analyst can specify the typed graph theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the graph class and finiteness, number and placement of cops, robber placement, turn order, legal moves and waiting, information and capture condition, winning strategy quantifiers and minimized cop count are explicit. The scope is broad within that domain but bounded by the need for the graph class and finiteness, number and placement of cops, robber placement, turn order, legal moves and waiting, information and capture condition, winning strategy quantifiers and minimized cop count are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the graph class and finiteness, number and placement of cops, robber placement, turn order, legal moves and waiting, information and capture condition, winning strategy quantifiers and minimized cop count 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 number. Cop number 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 theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the graph class and finiteness, number and placement of cops, robber placement, turn order, legal moves and waiting, information and capture condition, winning strategy quantifiers and minimized cop count are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of graph theory because they reuse the typed graph theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Cops choose initial vertices, the robber responds, players alternate along edges or wait and the least team size having a winning strategy over every robber response is assigned to the graph., and type the carrier, state every parameter and convention in the definition, test that the graph class and finiteness, number and placement of cops, robber placement, turn order, legal moves and waiting, information and capture condition, winning strategy quantifiers and minimized cop count are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Cop number Domain-specific
Parents (1) — more general patterns this builds on
-
Cop number is a kind of Game-Theoretic Strategy Prime
The proposed strict upward parent is
prime:game_theory_strategy.
Hierarchy path (1) — routes to 1 parentless root
- Cop number → Game-Theoretic Strategy → Function (Mapping)
Neighborhood in Abstraction Space¶
Cop number sits in a crowded region of the domain-specific corpus (5th 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-win graph — 0.95
- Entanglement (graph measure) — 0.95
- Eternal dominating set — 0.94
- Matching (graph theory) — 0.93
- Split graph — 0.93
Computed from structural-signature embeddings · 2026-09-08