### Constructible graphs and the game of cops and robbers

Combinatorics Seminar

4th October 2022, 11:00 am – 12:00 pm

Fry Building, 2.04

The game of cops and robbers is played on a fixed graph G. The cop picks a vertex to start at, and the robber then does the same. Then they move alternately, with the cop moving first: at each turn the player moves to an adjacent vertex or does not move. The game is won by the cop if he lands on the robber. The graph G is called cop-win if the cop has a winning strategy, and weak cop-win if the cop has a strategy that ensures that the robber is either caught or visits each vertex only finitely many times.

A graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood.

In this talk we focus on infinite graphs and the relation between constructible graphs and cop-win or weak cop-win graphs.

We also investigate how these notions relate to the (weaker and more natural) notion of ‘locally constructible’ (every finite subgraph is contained in a finite constructible subgraph).

Joint work with Imre Leader and Mark Walters.

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