Abstract
A motion of a robot in space is represented by a graph. A robot change its position from point to point and its position can be determined itself by distinct labelled landmarks points. The problem is to determine the minimum number of landmarks to find the unique position of the robot, this phenomena is known as metric dimension. Motivated by this a new modification was introduced by Kelenc. In this paper, we computed the edge metric dimension of barycentric subdivision of Cayley graphs Cay(Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub> ⊕Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sub> ), for every α ≥ 6, β ≥ 2 and an observation is made that it has constant edge metric dimension and only three carefully chosen vertices can appropriately suffice to resolve all the edges of barycentric subdivision of Cayley graphs Cay(Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub> ⊕ Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sub> ).
Highlights
Slater [1] and Harary el at [2] independently introduced the concept of metric dimension
Metric dimension has contributed in different real world applications like navigation of robots [3], wireless communications and sensor networks [4], pattern recognition and image processing [5], and above of all it has most applications in chemistry that are discussed in [6]–[8]
We proved that the metric dimension of the barycentric subdivision BS( ) of is constant and only three vertices appropriately chosen suffice to resolve all the vertices of the BS( )
Summary
Slater [1] and Harary el at [2] independently introduced the concept of metric dimension. Kelenc et al [20] explained a detailed comparison between the edge metric generator and standard metric generator In this paper, they showed the edim(G) is a NP-hard problem and determined that the edim(G) of grid graph is 2. Kelenc et al [20] proved that the edim(W1,n) of wheel graph W1,n is n for n = 3, 4 and n − 1 for n ≥ 5 They determined the edge metric dimension of path, cycle, complete graph, complete bipartite, Fan graphs, cartesian product of cycles and bounds for some families of graphs. We discussed the edge metric dimension of barycentric subdivision of Cayley graphs Cay(Zα ⊕ Zβ ). Ahmad: Barycentric Subdivision of Cayley Graphs With Constant Edge Metric Dimension
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