The fullerene graphs with a perfect star packing

Fullerene graph G is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. A spanning subgraph of G is called a perfect star packing in G if its each component is isomorphic to K-1,K- 3. For an independent set D subset of V(G), if each vertex in V(G) \ D has exactly one neighbor in D, then D is called an efficient dominating set of G. In this paper we show that the number of vertices of a fullerene graph admitting a perfect star packing must be divisible by 8. This answers an open problem asked by Doslic et al. and also shows that a fullerene graph with an efficient dominating set has 8n vertices. In addition, we find some counterexamples for the necessity of Theorem 14 of paper of Doslic et al. from 2020 and list some subgraphs that preclude the existence of a perfect star packing of type P0.