Automorphism Breaking of Finite and Infinite Graphs

This talk surveys results on automorphism breaking of graphs by vertex colorings. It briefly outlines several areas of research and open problems. In particular, it discusses automorphism breaking of vertex transitive graphs, tree-like graphs, graphs of bounded valence, the infinite motion conjecture for locally finite graphs, graphs of polynomial and subexponential growth, cost and density of 2-distinguishable graphs, products of graphs, uncountable graphs, and probabilistic results.