The purpose of this work is to determine the co-even domination number of various graphs, as a ladder, lollipop, butterfly, jellyfish, helm, corona, fan, and double fan graph. Before this, the important properties of the co-even dominating set are mentioned from previous work.

The aim of this article is to introduce a new definition of domination number in graphs called hn-domination number denoted by . This paper presents some properties which show the concepts of connected and independent hn-domination. Furthermore, some bounds of these parameters are determined, specifically, the impact on hn-domination parameter is studied thoroughly in this paper when a graph is modified by deleting or adding a vertex or deleting an edge.

The purpose of this paper is to introduce a new inverse domination parameter in the graphs it is called inverse co-even domination number. Some properties of the theory to this definition were only touched. Also, many properties and limitations on this definition are determined. Additionally, some properties of inverse co-even domination number for some certain graphs and its complement are founded, such as regular, complete, path, cycle, wheel, complete bipartite, and star.

In this paper, a new definition of graph domination called “Captive Domination” is introduced. The proper subset of the vertices of a graph [Formula: see text] is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which doesn’t belong to the dominating set. The inverse captive domination is also introduced. The lower and upper bounds for the number of edges of the graph are presented by using the captive domination number. Moreover, the lower and upper bounds for the captive domination number are found by using the number of vertices. The condition when the total domination and captive domination number are equal to two is discussed and obtained results. The captive domination in complement graphs is discussed. Finally, the captive dominating set of paths and cycles are determined.

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