Domination Numbers in Graphs Resulting from Shackle Operations with Linkage of any Graph

Abstract

The domination number is the number of dominating nodes in a graph that can dominate the surrounding connected nodes with a minimum number of dominating nodes. This domini number is denoted by γ(G). In this research, we will examine the domination number of the distance between two graphs resulting from the shackle operation with any graph as linkage. This differs from previous research, namely the domination of numbers at one and two distances. This study emphasizes how the results of operations on the shackle are connected to the shackle graph as any graph connects the copy. Any graph here means all graphs are connected and generally accepted. The method used in this research is pattern recognition and axiomatic deductive methods. The pattern detection method examines patterns where a graph's number of dominating points can dominate the connected points around it with a minimum number of dominating nodes. Meanwhile, axiomatic deductive is a research method that uses the principles of deductive proof that apply to mathematical logic by using existing axioms or theorems to solve a problem. The Result of graph S_n with t copies and S_m as linkage, then the two-distance domination number in the graph resulting from the shackle operation is γ_2 (Shack(S_n,S_m,t) )=t-1; graph S_n with t copies and C_m as linkage, then the two-distance domination number in the graph resulting from the shackle operation is γ_2 (Shack(S_n,C_m,t) )={■(t,for 3≤m≤6@⌈n/5⌉(t-1),for m≥7)┤; graph C_n with t copies and S_m as linkage, then the two-distance domination number in the graph resulting from the shackle operation isγ_2 (Shack(C_n,S_m,t) )={■(t-1,for n=3@t,for 4≤n≤5@⌈n/5⌉t,for n≥6)┤ This research provides benefits and adds to research results in the field of graph theory specialization of two-distance domination numbers in the result graph of shackle operation with linkage any graph.