Minimal Doubly Resolving Sets of Certain Families of Toeplitz Graph

Abstract

The doubly resolving sets are a natural tool to identify where diffusion occurs in a complicated network. Many real-world phenomena, such as rumour spreading on social networks, the spread of infectious diseases, and the spread of the virus on the internet, may be modelled using information diffusion in networks. It is obviously impractical to monitor every node due to cost and overhead limits because there are too many nodes in the network, some of which may be unable or unwilling to send information about their state. As a result, the source localization problem is to find the number of nodes in the network that best explains the observed diffusion. This problem can be successfully solved by using its relationship with the well-studied related minimal doubly resolving set problem, which minimizes the number of observers required for accurate detection. This paper aims to investigate the minimal doubly resolving set for certain families of Toeplitz graph T-n(1, t), for t >= 2 and n >= t + 2. We come to the conclusion that for T-n(1, 2), the metric and double metric dimensions are equal and for T-n(1, 4), the double metric dimension is exactly one more than the metric dimension. Also, the double metric dimension for T-n(1, 3) is equal to the metric dimension for n = 5, 6, 7 and one greater than the metric dimension for n >= 8.