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Aug 27, 2017 at 4:32 comment added Jimmy Dillies Let n(A) be the minimal number of components of a graph's Tait subgraphs, and let's denote by A' the vertex sum of three copies of A. Agol's answer shows that n(A')=3n(A)-2. By taking the iterated sequence A, A', (A')', ... one gets a family of planar cubic graphs with a strictly increasing number of components for its minimal Tait subgraph.
Aug 27, 2017 at 4:23 comment added Jimmy Dillies Ian Agol's gives a constructive proof:
Aug 26, 2017 at 20:56 history answered Peter Heinig CC BY-SA 3.0