Fault-tolerant resolvability of some graphs of convex polytopes

S. K. Sharma ; H. Raza ; V. K. Bhat

Diskretnaya Matematika, Tome 34 (2022) no. 4, pp. 108-122

Résumé

The fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set $\mathbb{W}$ of vertices of a connected graph $G$ in which the vector of distances to the vertices in $\mathbb{W}$ uniquely determines all the vertices of the graph $G$. The minimum cardinality of a resolving set of $G$ is known as the metric dimension of $G$. If $\mathbb{W}\setminus \rho$ is also a resolving set for each $\rho$ in $\mathbb{W}$. In that case, $\mathbb{W}$ is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set $\mathbb{W}$. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures.

Détail

Keywords: Convex polytopes, metric dimension, fault-tolerant metric dimension, connected graph, planar graph.