Computing Metric Dimension of Certain Families of Toeplitz Graphs

Abstract

The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , ... , q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> } be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), r(a, q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ), ... , r(a, q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> )), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> 〈1, t〉 and T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> 〈1, 2, t〉, respectively is discussed and proved that it is constant.