Constructible Graphs and Pursuit

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). Our other main aim is to investigate how this notion relates to the notion of `locally constructible' (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.