An Integrated Review on Graph Labeling and Domination Parameters in Graph Theory

In recent decades Graph labeling and Domination parameters have attracted consider- able attention from researchers due to their mathematical elegance and practical relevance. The research problems include to show how a star graph, splitting graph of a star graph and degree splitting graph of star K_1,n satisfies cordial graph property, further proving that a Jewel graph J_n, and jelly fish J_n,n also comes under cordial labeling. Moving onto the Product Cordial Labeling of graph, the problem includes researching the Product Cordial Labeling of a Cycle C_n, a complete graph K_n, K_m,n. Further research includes verifying the k-product cordial labeling of a new graph called cone graph constructed by joining each vertex of a cycle C_n and a pendant vertex K₁. Next, the study aims to investigate the harmonious labeling of graphs such as Petersen Graph, W_n. Moving further, it checks the odd and even harmonious labeling of Star Graphs and Cycles under different rules and situation. In another research paper, the researcher tries to analyze the graphs which shows G-Graceful Labeling. He chose path P_n, flower graph, Star graph, Bipartite graph and ladder along with different conditions to check if they are G- graceful graph. The problem includes how to use labeling for automatic routing of data in a network, how to use less memory or computer power and simultaneously helping data move faster across networks. They research how can labeling be used to study the structure of crystalline solid. The re- searchers formulated the problem to find Friendly Index Set of some classes of graphs under friendly labeling conditions and their vivid structural patterns. They formulated research problems to find the Divisor Cordial Labeling on the variants of a complete graph like the new graph obtained from a complete graph by dividing the set of vertices of G into union of sets which include at least 2 vertices from G. After the finding of Dominating set and Domination Number of graphs, the researchers formulated the problem to find the value of constant Cr in the formula of the domination number proved earlier. In the next research paper, the researcher investigates the domination number in context of graphs like the cycle with one chord, anti-prism graph and some other graphs. They further focused on finding the Maximal Dominating Cardinality and Maximum Dominating Cardinality for various graphs. With reference to dominating set, the new concept of fair domination, restrained domination and combining both, fair restrained domination was introduced and researchers proposed to construct and find new families of such graphs. Later on, the problem to char- acterize the Restrained Dominating Set on join of the graphs was formulated and could the outer restrained domination number of the combined graph be expressed exactly or bounded in terms of domination or restrained domination.