On the metric dimension of Möbius ladders (original) (raw)

Ars Combinatoria -Waterloo then Winnipeg-

Abstract

If G is a connected graph, the distance d(u,v) between two vertices u,v∈V(G) is the length of the shortest path between them. Let W={w 1 ,w 2 ,⋯,w k } be an ordered set of vertices of G and let v be a vertex of G. The representation r(v∣W) of v with respect to W is the k-tuple (d(v,w 1 ),d(v,w 2 ),⋯,d(v,w k )). If distinct vertices of G have distinct representations with respect to W, then W 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 family G of connected graphs is a family with constant metric dimension if dim(G) does not depend upon the choice of G in G. In this paper we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladders M n constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder M n except when n≡2(mod8). It is ...

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