Double total domination number in certain chemical graphs

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.

Keywords

• total domination
• double total domination
• hexagonal systems
• molecular graph
• pyrene network
• hexabenzocoronene