On the metric dimension of circulant graphs

Abstract Let G = ( V , E ) be a connected graph and d ( x , y ) be the distance between the vertices x and y in V ( G ) . A subset of vertices W = { w 1 , w 2 , … , w k } is called a resolving set or locating set for G if for every two distinct vertices x , y ∈ V ( G ) , there is a vertex w i ∈ W such that d ( x , w i ) ≠ d ( y , w i ) for i = 1 , 2 , … , k . A resolving set containing the 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 d i m ( G ) . Let F be a family of connected graphs G n : F = ( G n ) 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 d i m ( G n ) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension. The metric dimension of a class of circulant graphs C n ( 1 , 2 ) has been determined by Javaid and Rahim (2008) [13] . In this paper, we extend this study to an infinite class of circulant graphs C n ( 1 , 2 , 3 ) . We prove that the circulant graphs C n ( 1 , 2 , 3 ) have metric dimension equal to 4 for n ≡ 2 , 3 , 4 , 5 ( mod 6 ) . For n ≡ 0 ( mod 6 ) only 5 vertices appropriately chosen suffice to resolve all the vertices of C n ( 1 , 2 , 3 ) , thus implying that d i m ( C n ( 1 , 2 , 3 ) ) ≤ 5 except n ≡ 1 ( mod 6 ) when d i m ( C n ( 1 , 2 , 3 ) ) ≤ 6 .

[1]  Azriel Rosenfeld,et al.  Landmarks in Graphs , 1996, Discret. Appl. Math..

[2]  José Cáceres,et al.  On the metric dimension of some families of graphs , 2005, Electron. Notes Discret. Math..

[3]  András Sebö,et al.  On Metric Generators of Graphs , 2004, Math. Oper. Res..

[4]  David R. Wood,et al.  On the Metric Dimension of Cartesian Products of Graphs , 2005, SIAM J. Discret. Math..

[5]  Ioan Tomescu,et al.  On Metric and Partition Dimensions of Some Inflnite Regular Graphs , 2009 .

[6]  Gary Chartrand,et al.  On k-dimensional graphs and their bases , 2003, Period. Math. Hung..

[7]  M. Johnson,et al.  Structure-activity maps for visualizing the graph variables arising in drug design. , 1993, Journal of biopharmaceutical statistics.

[8]  Ioan Tomescu,et al.  Metric bases in digital geometry , 1984, Comput. Vis. Graph. Image Process..

[9]  Peter J. Cameron,et al.  Designs, graphs, codes, and their links , 1991 .

[10]  Azriel Rosenfeld,et al.  Localization in graphs , 1994 .

[11]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[12]  Gary Chartrand,et al.  Resolvability in graphs and the metric dimension of a graph , 2000, Discret. Appl. Math..

[13]  D. Frank Hsu,et al.  Distributed Loop Computer Networks: A Survey , 1995, J. Parallel Distributed Comput..