  We study the recognition complexity of subgraphs of 2- and 3-connected planar
cubic graphs. Recently, we presented [ESA 2022] a quadratic-time algorithm to
recognize subgraphs of planar cubic bridgeless (but not necessarily connected)
graphs, both in the variable and fixed embedding setting (the latter only for
2-connected inputs). Here, we extend our results in two directions: First, we
present a quartic-time algorithm to recognize subgraphs of 2-connected planar
cubic graphs in the fixed embedding setting, even for disconnected inputs.
Second, we prove NP-hardness of recognizing subgraphs of 3-connected planar
cubic graphs in the variable embedding setting.
