Double total domination number in certain chemical graphs

Ana Klobučar Barišić; Antoaneta Klobučar; Ana Klobučar Barišić; Antoaneta Klobučar

doi:10.3934/math.20221076

AIMS Mathematics

2022, Volume 7, Issue 11: 19629-19640. doi: 10.3934/math.20221076

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Research article

Double total domination number in certain chemical graphs

Ana Klobučar Barišić ^{1
,
,},
Antoaneta Klobučar ²

1.
Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia
2.
Faculty of Economics, Josip Juraj Strossmayer University of Osijek, Osijek, Croatia

Received: 13 July 2022 Revised: 24 August 2022 Accepted: 29 August 2022 Published: 06 September 2022
MSC : 05C69, 05C92

Let $ G $ be a graph with the vertex set $ V(G) $. A set $ D\subseteq V(G) $ is a total k-dominating set if every vertex $ v\in V(G) $ has at least $ k $ neighbours in $ D $. The total k-domination number $ \gamma_{kt}(G) $ is the cardinality of the smallest total k-dominating set. For $ k = 2 $ the total 2-dominating set is called double total dominating set. In this paper we determine the upper and lower bounds and some exact values for double total domination number on pyrene network $ PY(n) $, $ n\geq 1 $ and hexabenzocoronene $ XC(n) $ $ n\geq 2 $, where pyrene network and hexabenzocoronene are composed of congruent hexagons.
- total domination,
- double total domination,
- hexagonal systems,
- molecular graph,
- pyrene network,
- hexabenzocoronene
Citation: Ana Klobučar Barišić, Antoaneta Klobučar. Double total domination number in certain chemical graphs[J]. AIMS Mathematics, 2022, 7(11): 19629-19640. doi: 10.3934/math.20221076

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Abstract

Let $ G $ be a graph with the vertex set $ V(G) $. A set $ D\subseteq V(G) $ is a total k-dominating set if every vertex $ v\in V(G) $ has at least $ k $ neighbours in $ D $. The total k-domination number $ \gamma_{kt}(G) $ is the cardinality of the smallest total k-dominating set. For $ k = 2 $ the total 2-dominating set is called double total dominating set. In this paper we determine the upper and lower bounds and some exact values for double total domination number on pyrene network $ PY(n) $, $ n\geq 1 $ and hexabenzocoronene $ XC(n) $ $ n\geq 2 $, where pyrene network and hexabenzocoronene are composed of congruent hexagons.

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