Recognition Complexity of Subgraphs of 2- and 3-Connected Planar Cubic Graphs

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.