Power Dominating Numbers In Graphs - Chan, Kar Man (2021)
A Phase Measurement Unit (PMU) is a device to monitor the electrical activity and every electrical company uses it. Since PMU comes at a high cost and the company wants to use the least amount of PMU while monitoring all the electrical network stations to make sure they could respond to any emergency situation. When we convert this problem into Graph Theory, we have the power dominating problem which is to find the minimum cardinality of the smallest power dominating set (PDS) of a graph (i.e. the power dominating number). In this project, we will be investigating the power dominating number of a specific graph called twisted torus, which is a variation of torus graph. To find the power dominating number, we have to understand the observation rules and apply it properly. Then, we have to study and analyze the power dominating problem for various graphs. For example, the torus and the cylinder graph have the closest resemblance of a twisted torus. Once we have gone through that, we will begin the first phase of the proof. That is, find the zero forcing number for the twisted torus such that we could apply it to a known theorem in order to find the lower bound of the power dominating number. To find the zero forcing number, we have broken down the problem into different parts in order to get a good grasp on it. If the zero forcing number is found, we may enter the second and the last phase of the proof. That is, find the lower bound and the upper bound of power dominating number of the twisted torus. In this phase, we will show the construction of the PDS and the bounded region of the power dominating number of the twisted torus.