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Mathematical Sciences Letters
An International Journal


Volumes > Vol. 5 > No. 1


Metric Dimension of Some Families of Graph

PP: 99-102
Gohar Ali, Roohi Laila, Murtaza Ali,
If G is a connected graph, the distance d(x, y) between two vertices x, y ∈ V(G) is the length of a shortest path between them. Let d(x, y) denote the distance between vertices x and y of a connected graph G. If d(z, x) 6= d(z, y), then z is said to resolve x and y and therefore z is called a resolving vertex for the vertices x and y. LetW = {w1,w2, . . . ,wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v|G) of v with respect to W is the k-tuple (d(v,w1),d(v,w2), . . . ,d(v,wk)). If distinct vertices of G have distinct representations with respect toW, thenW is called a resolving set or locating set for G. A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension of G, denoted by dim(G). A familyG of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G . In this paper, we find the constant metric dimension of Pn(1,2,3) and Mn.

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