Barycentric Subdivision of Cayley Graphs With Constant Edge Metric Dimension
Ali N. A. Koam, Ali Ahmad
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> ).