On resolvability and exchange property in antiweb-wheels
Abstract
Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn:F - (Gn) n≥1 depending on n as follows: the order |V(G)| = ℓ(n) and lim ℓ(n) = ∞. If there exists a constant C > 0 such that β(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise T has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), F is called a family with constant metric dimension. In this paper, we study the metric dimension of antiweb-wheels. We determine the exact value of metric dimension for antiweb-wheels and prove that they have unbounded metric dimension. It is natural to ask for characterization of graphs classes with respect to the nature of their metric dimension. It is also shown that the exchange property of the bases in a vector space does not hold for minimal resolving sets of antiweb-wheels.
