Locally finite graphs and their localization numbers
Abstract
We study the Localization Game on infinite trees, where each of the countably many vertices has finite degree. In contrast to the finite case, we construct a locally finite tree with localization number n for any choice of positive integer n. Our examples have uncountably many ends, and we show that this is necessary by proving that locally finite trees with finitely or countably many ends have localization number at most 2. Finally, as is the case for finite graphs, we prove that any locally finite graph contains a subdivision where one cop can capture the robber.
Fields of Study & Subjects
Advanced Graph Theory Research
Graph theory and applications
Graph Labeling and Dimension Problems
Publication details
- Journal / source
- Journal of Combinatorics
- Publisher
- International Press of Boston
- Volume
- 17
- Issue
- 4
- Pages
- 467–481
- Publication date
- 1 January 2026
- Language
- EN