Cop-win graph¶

A graph on which one pursuer has a strategy that guarantees capture of one evader in the alternating vertex-movement game.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3909
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Graph Pursuit Games → Mathematics
Aliases: Dismantlable graph, Constructible graph

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.

How would you explain it like I'm…

One Cop Always Wins

Imagine a map of dots joined by roads, with one cop and one robber taking turns moving to a next-door dot or staying still. A cop-win graph is a map where just one cop can always catch the robber, no matter how the robber runs.

Maps Where One Cop Wins

A cop-win graph is a map of spots and paths where a single cop is always able to catch a robber. The cop picks a starting spot, then the robber does, and they take turns moving one path or staying still. There's a neat way to test a map: look for a spot that is 'covered': some other spot is right next to it and also next to every one of its neighbors. If you can keep removing covered spots one at a time until only one spot is left, then one cop can always win.

Dismantlable Cop-Win Graphs

A cop-win graph is a graph on which a single cop can always catch the robber in the Cops and Robbers game. The cop chooses a starting vertex, the robber responds, and then they alternate, each moving along an edge or staying put. For finite graphs where every vertex counts as next to itself, there is a neat test called dismantlability: repeatedly remove a vertex whose neighborhood (including itself) is contained in another vertex's neighborhood; the graph is cop-win exactly when this can continue down to a single vertex. The idea is that a removed vertex is 'covered' by another one, so the cop can treat the robber on it as if it were on the covering vertex. The exact rules, such as who places first and whether staying put is allowed, must be stated.

A cop-win graph is one whose cop number is 1: a single cop has a winning strategy against the robber. The convention is a finite undirected reflexive graph (every vertex has a loop, so waiting is a legal move); the cop chooses first, the robber responds, and each then moves along an edge or remains. The characterization is dismantlability: repeatedly remove a vertex whose closed neighborhood is contained in another vertex's closed neighborhood, and the graph is cop-win exactly when this process can reduce it to a single vertex. The proof uses the domination to define a retraction from the graph onto the smaller one: the cop plays against the robber's image on the smaller graph and, having caught that shadow, finishes the capture on the original. The result depends on the stated assumptions — finiteness, the loop convention, one cop, initial-choice order, full information and the capture rule — and changing them changes the class.

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.
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) — outgoing typed relations

Cop-win graph is a kind of Coverage / Reachability Prime

It remains its own entry because its identity is fixed by 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.

Hierarchy paths (2) — routes to 2 parentless roots

Cop-win graph → is a kind of Coverage / Reachability → is a kind of Completeness

Show alternative path (1)

Neighborhood in Abstraction Space¶

Cop-win graph sits in a crowded region of the domain-specific corpus (21^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Pursuit Games on Graphs (5 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25