A review on domination number of graphs

Domination has its origin in 1862 when C. F. De Jaenisch investigated the issue of identifying the smallest number of queens required to cover a chessboard. Around 1960 Berge and Ore started the mathematical exploration of domination theory in graphs. Let Z = (X, Y) denote any finite graph, and D represents any subset of vertex set X. A set of vertices D in a graph Z = (X, Y), is said to be a dominating set(DS) if the neighborhood of D is whole of Xz i. e., Nz[D] = Xz. The minimal cardinality of DS for Z is, therefore, the domination number (DN) denoted by γ(Z). In this paper we have reviewed literature on domination theory of graphs and then investigated and generalized some general results for DS and DN of the graphs such as diamond snake(Dm), banana tree(B(m, n)), coconut tree(CT(m, n)), and firecracker(F(m, n)).