Constructible Graphs and Pursuit
Maria-Romina Ivan, Imre Leader, Mark Walters

TL;DR
This paper explores the properties of constructible graphs, providing new examples and answering open questions about their structure, rank, and relation to cop-win and locally constructible graphs, with implications for graph pursuit games.
Contribution
It presents the first known cop-win but not constructible graph, shows every countable ordinal as a graph rank, and investigates the relationship between locally constructible and weak cop-win graphs.
Findings
Constructible graphs include all finite cop-win graphs.
A cop-win but non-constructible graph is constructed.
Every countable ordinal can be realized as the rank of a constructible graph.
Abstract
A (finite or infinite) 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. One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism, answering a question of Chastand, Laviolette and Polat. Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set).…
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