Abstract
Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension.
Highlights
Resolvability of graphs becomes an important parameter in graph theory due to its wide applications in different branches of mathematics, such as facility location problems, chemistry, especially molecular chemistry [1], the method of positioning robot networks [2], the optimization problem in combinatorics [3], applications in pattern recognition and image processing [4], and the problems of sonar and Coast Guard LORAN [5]
E resolvability of graphs depends on the distances in graphs. e distance between two vertices in a connected graph is the smallest distance connecting those two vertices. e representation of a vertex u with respect to the set W is denoted by r(u, W) and is defined as a k-tupple (d(u,w1), \dots, d(u,wn)), where w1, \dots, wn \in W. e set W is called the resolving set [1] or sometimes locating set [5] if each vertex of the graph has a unique representation with respect to W
A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph, generally denoted by β(G)
Summary
Resolvability of graphs becomes an important parameter in graph theory due to its wide applications in different branches of mathematics, such as facility location problems, chemistry, especially molecular chemistry [1], the method of positioning robot networks [2], the optimization problem in combinatorics [3], applications in pattern recognition and image processing [4], and the problems of sonar and Coast Guard LORAN [5].e resolvability of graphs depends on the distances in graphs. e distance between two vertices in a connected graph is the smallest distance connecting those two vertices. e representation of a vertex u with respect to the set W is denoted by r(u, W) and is defined as a k-tupple (d(u,w1), \dots, d(u,wn)), where w1, \dots, wn \in W. e set W is called the resolving set [1] or sometimes locating set [5] if each vertex of the graph has a unique representation with respect to W. It is enough to show that every vertex of the graph Gn[1, 4] is uniquely determined by some vertices in W. Is shows that every vertex of the graph Gn[1, 4] is uniquely determined by some of the vertices in W.
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