A Mathematical Study of Some Aspects of Split Domination Number of Graph Towards Generalisation

Abstract

In this thesis, we have presented the generalized formulae for determining the split newlinedomination number of the line graph and dual graph of the specific classes of graphs. newlineAdditionally, an algorithm has been developed to compute the split domination newlinenumber of these defined classes of graphs, along with exploring the practical newlineapplications within various domains. newlineOur work commences with the reverse engineering process of analysing practical newlineapplications such as computer networking, public networks, optimization theory, newlinecombinatorics, social networks, electrical networks, and various chemical newlinecompounds. Through the examination of these networks, we recognize the newlinesignificance of domination. It becomes apparent to us that certain nodes are essential newlinein dominating the network from different perspectives. Dominating sets can help in newlinethe identification of key nodes, such as routers or switches that have the ability to newlineoversee or regulate the entire network, ensuring efficient resource utilization while newlinemaintaining network connectivity. In the realm of public transportation, it is crucial newlineto identify bus stops or train stations (nodes) that can efficiently serve a maximum newlinenumber of passengers (coverage) with minimal stops. Additionally, recognizing key newlineinfluencers (dominating nodes) who can effectively disseminate information or newlineinfluence behaviour throughout the network is essential. Moreover, finding minimal newlinesets of individuals who can successfully monitor or distribute information within newlinecommunities is also vital. Lastly, fault analysis is imperative in the event of newlineelectrical network failures. While studying the network from domination point of newlineview it is noted by us, the importance of splitting of network along with domination, newlineso we point out such graph and do study of splitting of network by focusing on newlinesplitting using connections and splitting using regions. After outlining the necessary tasks based on a review of literature and identifying newlineresearch gaps in chapters 1 and 2, the third chapter presents new formulae