Domination and independent domination in Hexagonal systems

Journal article

Authors/Editors

PAWATON KAEMAWICHANURAT

Strategic Research Themes

Modeling Design and Optimization (Computational Science and Engineering)

Publication Details

Author list: Almalki N., Kaemawichanurat P.

Publication year: 2022

Volume number: 10

Issue number: 1

Start page: 1

End page: 22

Number of pages: 22

ISSN: 22277390

URL: https://www.scopus.com/inward/record.uri?eid=2-s2.0-85121768205&doi=10.3390%2fmath10010067&partnerID=40&md5=ca32b23c142c522dba9257b4e6bfb064

Languages: English-Great Britain (EN-GB)

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Abstract

A vertex subset D of G is a dominating set if every vertex in V(G) \ D is adjacent to a vertex in D. A dominating set D is independent if G[D], the subgraph of G induced by D, contains no edge. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G, and the independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. A classical work related to the relationship between γ(G) and i(G) of a graph G was established in 1978 by Allan and Laskar. They proved that every K1,3-free graph G satisfies γ(G) = i(H). Hexagonal systems (2 connected planar graphs whose interior faces are all hexagons) have been extensively studied as they are used to present bezenoid hydrocarbon structures which play an important role in organic chemistry. The domination numbers of hexagonal systems have been studied continuously since 2018 when Hutchinson et al. posted conjectures, generated from a computer program called Conjecturing, related to the domination numbers of hexagonal systems. Very recently in 2021, Bermudo et al. answered all of these conjectures. In this paper, we extend these studies by considering the relationship between the domination number and the independent domination number of hexagonal systems. Although every hexagonal system H with at least two hexagons contains K1,3 as an induced subgraph, we find many classes of hexagonal systems whose domination number is equal to an independent domination number. However, we establish the existence of a hexagonal system H such that γ(H) < i(H) with the prescribed number of hexagons. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

Keywords

Hexagonal systems, Independent domination