Before revision:  It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and planar graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.
After revision:  It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and H-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.