Total and Double Total Domination Number on Hexagonal Grid

Klobučar, Antoaneta; Klobučar, Ana

doi:10.3390/math7111110

Cited by 10 publications

(5 citation statements)

References 5 publications

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“…Currently, there are only a limited number of publications on total and double total domination on chemical graphs [2,6,10,12,14,15]. This work is in a close relationship with our previous papers [10,12], in which we also study double total domination, but on a hexagonal grid and pyrene network.…”

Section: Introductionsupporting

confidence: 75%

Double Total Domination Number on Some Chemical Nanotubes

KLOBUČAR BARIŠIĆ,

KLOBUČAR

2024

KgJMat

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Suppose G is a graph with the vertex set V (G). A set D ⊆ V (G) is a total k-dominating set if every vertex v ∈ V (G) has at least k neighbours in D. The total k-domination number γkt(G) is the size of the smallest total k-dominating set. When k = 2 the total 2-dominating set is referred to as a double total dominating set. In this work we compute the exact values for double total domination number on H-phenylenic nanotubes HPH(m,n), m,n ≥ 2 and H-naphtalenic nanotubes HN(m,n), n = 2k, m,n ≥ 2. As all vertices have a degree 2 or 3, there is no total k-domination for k ≥ 3 for H-phenylenic and H-naphtalenic nanotubes, and the double total domination is the maximum possible.

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Section: Introductionsupporting

confidence: 75%

Double Total Domination Number on Some Chemical Nanotubes

KLOBUČAR BARIŠIĆ,

KLOBUČAR

2024

KgJMat

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“…In the previous period, t-dominance was investigated on hexagonal [4][5][6][7], rectangular [8], and triangular [9] cactus chains. Research was also conducted for R-domination in graphs [10], k-dominating number of Cartesian products of two paths [11], paid domination in graphs [12], and total and double-total domination number on the hexagonal network [13].…”

Section: Methodsmentioning

confidence: 99%

Dominating Number on Icosahedral-Hexagonal Network

Carević¹

2021

Mathematical Problems in Engineering

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“…For pericondensed hexagonal systems, we focus on the following well-known classes. The first class of pericondensed hexagonal systems was introduced by Klobučar and Klobučar [12].…”

Section: Pericondensed Hexagonal Systemsmentioning

confidence: 99%

Domination and Independent Domination in Hexagonal Systems

Almalki

Kaemawichanurat

2021

Mathematics

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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.

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Total and Double Total Domination Number on Hexagonal Grid

Cited by 10 publications

References 5 publications

Double Total Domination Number on Some Chemical Nanotubes

Double Total Domination Number on Some Chemical Nanotubes

Dominating Number on Icosahedral-Hexagonal Network

Domination and Independent Domination in Hexagonal Systems

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