Additional Properties of Frame Domination in Graphs

Abstract

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.

Examining Captive and Inverse Captive Domination in Selected Graphs and Their Complements

“…The domination number 𝛾(𝐺) is the minimum cardinality of a dominating set D of G. If V-D contains a dominating set, then this set is called an inverse set of D in G. The symbol 𝛾 −1 (𝐺) represents the minimum cardinality over all inverse dominating set of G [2]. The concept of domination is used to solve many problems in various fields of mathematics subjects such as topological graphs [2,3], fuzzy graphs [4,5], and [6,7], number theory graph [8], general graphs [9][10][11][12][13][14][15][16][17][18], and others. The reader can be found all notions not mentioned [19][20][21][22].…”