On Cycle Related Graphs with Constant Metric Dimension ()

Murtaza Ali, Gohar Ali, Usman Ali, M. T. Rahim

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**DOI: **10.4236/ojdm.2012.21005
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If G is a connected graph, the distance d (u,v) between two vertices u,v ∈ V(G) is the length of a shortest path between them. Let W = {w_{1}, w_{2}, ..., w_{k}} be an ordered set of vertices of G and let v be a vertex of G . The repre-sentation r(v|W) of v with respect to W is the k-tuple (d(v,w_{1}), d(v,w_{2}), …, d(v,w_{k})). . If distinct vertices of G have distinct representations with respect to W , then W is called a resolving set or locating set for G. A re-solving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension of G , denoted by dim (G). A family ? of connected graphs is a family with constant metric dimension if dim (G) is finite and does not depend upon the choice of G in ?. In this paper, we show that dragon graph denoted by T_{n,m} and the graph obtained from prism denoted by 2C_{k} + {x_{k}y_{k}} have constant metric dimension.

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M. Ali, G. Ali, U. Ali and M. Rahim, "On Cycle Related Graphs with Constant Metric Dimension," *Open Journal of Discrete Mathematics*, Vol. 2 No. 1, 2012, pp. 21-23. doi: 10.4236/ojdm.2012.21005.

Conflicts of Interest

The authors declare no conflicts of interest.

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