We study the recognition complexity of subgraphs of k-connected planar cubic graphs for k = 1, 2, 3. We present polynomial-time algorithms to recognize subgraphs of 1- and 2-connected planar cubic graphs, both in the variable and fixed embedding setting. The main tools involve the Generalized (Anti)factor-problem for the fixed embedding case, and SPQR-trees for the variable embedding case. Secondly, we prove NP-hardness of recognizing subgraphs of 3-connected planar cubic graphs in the variable embedding setting.

翻译：我们研究了k=1,2,3时k-连通平面三次图子图的识别复杂性。我们提出了多项式时间算法，用于在可变嵌入和固定嵌入设置下识别1-连通和2-连通平面三次图的子图。主要工具包括固定嵌入情况下的广义（反）因子问题，以及可变嵌入情况下的SPQR树。其次，我们证明了在可变嵌入设置下识别3-连通平面三次图子图是NP困难的。