Abstract
In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid H m , n with m hexagons in a row and n hexagons in a column. Further, we explore the ratio between the total domination number and the number of vertices of H m , n when m and n tend to infinity.
Highlights
Graph dominations are widely applied in different problems such as dominating queens, computer network, school bus routing, and social network problems
We explore the ratio between the total domination number and the number of vertices of Hm,n when m and n tend to infinity
We determined the upper and lower bounds for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid Hm,n with m hexagons in a row and n hexagons in a column
Summary
Graph dominations are widely applied in different problems such as dominating queens, computer network, school bus routing, and social network problems. Hexagonal systems are geometric objects that are obtained by arranging congruent regular hexagons in a plane They are of significant importance in theoretical chemistry as a natural graph representation of benzenoid hydrocarbons [1,5,11]. We explore the ratio between the total domination number and the number of vertices of Hm,n when m and n tend to infinity At this moment, there are only few publications on total and double-total domination on hexagonal chains [3,12], but none dealing with arbitrary grids. There are only few publications on total and double-total domination on hexagonal chains [3,12], but none dealing with arbitrary grids Apart from this Introduction, the rest of the paper is organized as follows.
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