A survey and classification of Sierpiński-type graphs

The purpose of this survey is to bring some order into the growing literature on a type of graphs which emerged in the past couple of decades under a wealth of names and in various disguises in different fields of mathematics and its applications. The central role is played by Sierpiński graphs, but we will also shed some light on variants of these graphs and in particular propose their classification. Concentrating on Sierpiński graphs proper we present results on their metric aspects, domination-type invariants with an emphasis on perfect codes, different colorings, and embeddings into other graphs.