Packing stars in fullerenes

Scientific paper - Original scientific paper

• Author: Tomislav Došlić
• Author: Meysam Taheri-Dehkordi
• Author: Gholam Hossein Fath-Tabar
• Author's institution: University of Zagreb; Faculty of Civil Engineering
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract (english)

A spanning subgraph of a graph G is called a perfect star packing in G if every component
of the spanning subgraph is isomorphic to the star graph K1,3. An efficient dominating set
of graph G is a vertex subset D of G such that each vertex of G not in D is adjacent to
exactly one vertex from D and any two vertices of D are not adjacent in G. Fullerene graph
is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the
molecular graph of carbon fullerene. Clearly, a perfect star packing in a fullerene graph
G on n vertices will exist if and only if G has an efficient dominating set of cardinality n 4
. The problem of finding an efficient dominating set is algorithmically hard [2]. In this
paper, we give a characterization for a fullerene graph to own a perfect star packing. And
mainly show that it is necessary for a fullerene G owning a perfect star packing to have
order being divisible by 8. This answers an open problem asked by Dosli´c et. al. and also
shows that a fullerene graph with an efficient dominating set has 8n vertices. By the way, we
find some counterexamples for the necessity of Theorem 14 in [13] and list some forbidden
configurations to preclude the existence of a perfect star packing of type P0

• Keywords (english): Fullerene graph; Perfect star packing; Efficient dominating set
• Language: english
• Publication type: Scientific paper - Original scientific paper
• Publication status: Published
• Peer review: Peer review - international
• Publication version: Published version
• Journal title: Journal of Mathematical Chemistry
• Numbering: vol. 58, pp. 2223-2244
• p-ISSN: 0259-9791
• e-ISSN: 1572-8897
• URN:NBN: urn:nbn:hr:237:675944
• Publication: 2020
• Type of resource: Text
• Access conditions: Open access
• Created on: 2024-09-04 09:55:09