Abstract
Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.
Highlights
In a connected graph G (V, E), where V is the set of vertices and E is the set of edges, the distance d(u, v) between two vertices u, v ∈ V is the length of shortest path between them
A resolving set of minimum cardinality is called a basis for G and cardinality is the metric dimension of G, denoted by dim( G ) [3]
The concept of resolving set and metric basis have previously appeared in the literature [4,5,6]
Summary
In a connected graph G (V, E), where V is the set of vertices and E is the set of edges, the distance d(u, v) between two vertices u, v ∈ V is the length of shortest path between them. The representation r (v|W ) of v with respect to W is the k-tuple (d(v, w1 ), d(v, w2 ), d(v, w3 ), ..., d(v, wk )}, where W is called a resolving set [1] or locating set [2] if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. Motivated by the problem of uniquely determining the location of an intruder in a network, the concept of metric dimension was introduced by Slater in [2,7] and studied independently by Harary and Melter in [5]. Our main aim of this paper is to compute the metric dimension of graphs obtained from the rooted product graphs For this purpose, we need the following definitions.
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