On large light graphs in families of polyhedral graphs

A graph H is said to be light in a family H of graphs if each graph G∈H containing a subgraph isomorphic to H contains also an isomorphic copy of H such that each its vertex has the degree (in G) bounded above by a finite number φ(H,H) depending only on H and H. We prove that in the family of all 3-connected plane graphs of minimum degree 5 (or minimum face size 5, respectively), the paths with certain small graphs attached to one of its ends are light.