Abstract A graph G is 2-stratified if its vertex set is partitioned into two nonempty classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v . The F -domination number of a graph G is the minimum number of red vertices of G in a red–blue coloring of the vertices of G such that for every blue vertex v of G , there is a copy of F in G rooted at v . In this paper, we survey recent results on the F -domination number for various 2-stratified graphs F .