# On zig-zag chain graphs generated by regular hexagons with unbounded metric dimension

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## Abstract

Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W = {w1,w2,...,wk} is called a resolving set for G if for every two distinct vertices x,y ∈ V(G), there is a vertex wi ∈ W such that d(x,wi) ≠ d(y,wi). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension denoted by dim(G). Let F be a family of connected graphs Gn: F = (Gn)n≥1 depending on n as follows: the order | V(G) | = φ(n) and lim n→ ∞ φ(n) = ∞. If there exists a constant C > 0 such that dim(G) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension; otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), then F is called a family with constant metric dimension. In this paper, we study the metric dimension of some zig-zag chain graphs generated by regular hexagons. We determine the exact value of their metric dimension and prove that these zig-zag chain graphs have unbounded metric dimension. It is an interesting and classical problem to classify the graphs families with respect to the nature of their metric dimension.

Original language English 411-421 11 Utilitas Mathematica 100 Published - 2016 Yes

## Keywords

• Basis
• Metric dimension
• Resolving set
• Unbounded
• Zig-zag chain graphs

## ASJC Scopus subject areas

• Statistics and Probability
• Statistics, Probability and Uncertainty
• Applied Mathematics

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