Forbidden quadruplets generating a finite set of 2-connected graphs

For a graph GG and a set HH of connected graphs, GG is said to be HH-free if GG does not contain any member of HH as an induced subgraph. We let G2(H)G2(H) denote the set of all 2-connected HH-free graphs. Fujisawa et al. (2013) characterized sets HH of connected graphs such that |H|≤3|H|≤3 and G2(H)G2(H) is finite. In this paper, we extend their result and characterize sets HH of connected graphs such that |H|≤4|H|≤4 and G2(H)G2(H) is finite.