A study on the domforcing numbers of graphs

Abstract

Graph theory has wide-ranging applications across science, engineering, andreal-world systems. It studies abstract structures used to model relationshipsamong objects. Significant attention has been given to vertex parameters suchas domination, zero forcing number, k-forcing number, and their connected vari-ants in recent years. This thesis introduces a new concept, described as dom-forcing set and links it to connectedness, propagation time and k-forcing sets.Domination and zero forcing are two fundamental and well-studied concepts ingraph theory, each with important applications in areas such as network analysis,quantum system control, power grid monitoring, and information dissemination.The notion of a dom-forcing set naturally arises from combining these ideas. Itrepresents a set of vertices that not only dominate the graph locally but also ini-tiate a forcing process that eventually influences the entire graph. This synthesisprovides a powerful framework for understanding how control and influence prop-agate in complex networks. The study develops these concepts by establishingbounds, determining exact values for specific classes of graphs, and presentingvarious characterizations. A dominating set Df ⊆ V (Γ) in a graph Γ is calleda dom-forcing set if the subset Df forms a zero forcing set. The minimum car-dinality of such a set is defined as the dom-forcing number of Γ, denoted byFd (Γ).The thesis is organized into eleven chapters. Chapters 1 and 2 present thebroad context and necessary preliminaries. Chapters 3, 4, and 5 focus on thedom-forcing number, its behavior under different graph operations, and dom-forcing propagation time. Connected dom-forcing sets are examined in Chapters6 and7. Chapters 8 and 9 extend these ideas through generalizations to dom-k-forcing and connected dom-k-forcing sets. The thesis concludes in Chapter10, followed by directions for future research in Chapter 11. Overall, this workcontributes to the growing body of research in graph theory by introducing andsystematically analyzing dom-forcing related parameters. The results not onlydeepen the theoretical understanding of domination and zero forcing processesbut also open new directions for further investigation, particularly in the studyof dynamic processes on networks. The concepts and techniques developed inthis thesis are expected to have potential applications in areas such as networkcontrol, information propagation, and complex system analysis.