RESUMO
The concept of a metric dimension was proposed to model robot navigation where the places of navigating agents can change among nodes. The metric dimension m d ( G ) of a graph G is the smallest number k for which G contains a vertex set W, such that | W | = k and every pair of vertices of G possess different distances to at least one vertex in W. In this paper, we demonstrate that m d ( H D N 1 ( n ) ) = 4 for n ≥ 2 . This indicates that in these types of hex derived sensor networks, the least number of nodes needed for locating any other node is four.
RESUMO
The concept of resolving set and metric basis has been very successful because of multi-purpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called a fault-tolerant self-stable system. Recent research studies reveal that the problem of finding metric dimension is NP-hard for general graphs and the problem of computing the exact values of fault-tolerant metric dimension seems to be even harder although some bounds can be computed rather easily. In this article, we compute closed formulas for the fault-tolerant metric dimension of lattices of two types of boron nanotubes, namely triangular and alpha boron. These lattices are formed by cutting the tubes vertically. We conclude that both tubes have constant fault tolerance metric dimension 4.