Light subgraphs in planar graphs of minimum degree 4 and edge-degree 9

Let G be the class of simple planar graphs of minimum degree ≥ 4 in which no two vertices of degree 4 are adjacent. A graph H is light in G if there is a constant w such that every graph in G which has a subgraph isomorphic to H also has a subgraph isomorphic to H whose sum of degrees in G is ≤ w. Then we also write w(H) ≤ w. It is proved that the cycle Cs is light if and only if 3 ≤ s ≤ 6, where w(C3) = 21 and w(C4) ≤ 35. The 4-cycle with one diagonal is not light in G, but it is light in the subclass of all triangulations. The star K1,s is light if and only if s ≤ 4. In particular, w(K1,3) = 23. The paths Ps (s ≥ 1) are light, and w(P3) = 17 and w(P4) = 23.