Some properties of various dominations in graphs

Graph theory is considered a new language for all sciences such as medicine, engineering, economics, chemistry, physics, and others. The domination notion (the dominating set and the domination number) is currently considered one of the most used concepts in graph theory to find solutions to most of life's problems. Therefore, many definitions of this concept appeared, and each definition is an integrated work system that works to find the number of domination under certain conditions imposed by the type of problem. Throughout this study, the Hn-domination is discussed. Moreover, the following graphs, that have real-life applications, are defined: ladder graph, helm graph, tadpole graph, prism graph, sun graph, barbell graph, lollipop graph, triangular snake and double triangular snake graphs, windmill and Dutch windmill graphs, and caterpillar tree. For each of these graphs, each of the Hn-domination and inverse Hn-domination are determined. Furthermore, for the complements of the aforementioned graphs, each of the Hn-domination and inverse Hn-domination are also determined.