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The generalization of frame domination concept on graphs is the main aims of this work. In addition, some advanced properties on frame domination have been presented. Some corresponding theorems related to delete, add edges or delete vertices have been proved. Also, the relationship between the original graph and a graph getting from contraction edges has been addressed. Finally, the upper and lower bounds of frame domination number for some special different graphs have been compared and determined.

The generalization of the concept of inverse frame domination on graphs is the fundamental point of this work. It has been shown to present modern properties on inverse frame domination and some corresponding theorems related to delete, add edges or remove vertices. The relationship between the initial graph and a graph obtaining from the edges of the contraction was also discussed.

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