Total Face Irregularity Strength of Certain Graphs
Algorithm 4
Total face irregularity strength of sibling tree.
Input: Sibling tree ,.
Algorithm:
Step 1: Let denote a cycle of length three which is formed in Level x, for and the number of copies, represented by for . The root vertices of each level of is called as the pivot vertices and the sum of the weights of is called as the weight of . Notice that there will be copies of in each Level x.
Step 2: Let us begin by defining a rule for the labeling process. In order to label , we retain the labeling of and after retaining the labeling of , we start labeling the remaining copies of . In , there will be three copies of and we label the vertices and edges of by 1. Thus, the weight of is 6. Similarly, the vertices and edges of and will be labeled as follows. The vertices and edges of copies of are labeled temporarily as they were for , except for the pivot vertices. Observe that the two copies of will have the same weight as . In order to make the weights distinct, we change the label of the left non-pivot vertex of to 2; and in the rightmost copy of , we change the label of two non-pivot vertices to 2. Thus, the weight of the and are 7 and 8, respectively. Now we can say that copies of are permanently labeled. We can now begin labeling the vertices and edges of , for from Step .
Step 3: When , the vertices and edges in of are labeled temporarily as they were for , except for the pivot vertices. Observe that, there will arise two cases.Case (i):When the labels of the pivot vertices of , then the weights, .Case (ii):When the labels of the pivot vertices of or , then the weights, .Hence satisfies Case and the weights of and are all distinct.Now the vertices and edges of are labeled temporarily as they were for , except for the pivot vertices. Observe that, there will arise another case.Case (iii):When the labels of the pivot vertices of or , then the weights, .
