On large light graphs in families of polyhedral graphs

A graph HH is said to be light in a family HH of graphs if each graph G∈HG∈H containing a subgraph isomorphic to HH contains also an isomorphic copy of HH such that each its vertex has the degree (in GG) bounded above by a finite number φ(H,H)φ(H,H) depending only on HH and HH. 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.