Fractional Strong Metric Dimension of Convex Polytopes and its applications

Abstract

The fractional versions of various metric related parameters have recently gained importance due to their applications in the fields of sensor networking, robot navigation and linear optimization problems. Convex polytopes are collection of those polytopes of Euclidean space which are their convex subsets. They have key importance in the field of network designing due to their stable and resilient structure which aids optimal data transfer. The identification and removal of components (nodes) of a communication network causing abruption in its flow is of key importance for optimal data transmission. These components are referred as strong resolving neighbourhood (SRNs) in graph theory and assigning least weight to these components aids the computation of fractional strong metric dimension (FSMD). In this paper, we compute FSMD for certain convex polytopes which include $\mathbb{P}_{n}$, $\mathbb{P}_{n}^{1}$ and $\mathbb{P}_{n}^{2}$. In this regard, it is shown that for $n \geq 3$, FSMD of $\mathbb{P}_{n}$ and $\mathbb{P}_{n}^{2}$ is $n$ and $\frac{3n}{2}$, respectively. Also, FSMD of $\mathbb{P}_{n}^{1}$ is $n$ when $n$ is odd and $\frac{3n}{2}$ when $n$ is even. Finally, an application of FSMD in the context of internet connection networks is furnished.