Abstract
If G is a connected graph, the distance d(u,v) between two vertices u,v ε V(G) is the length of a shortest path between them. Let W = {w 1,2, ⋯,Wk} be an ordered set of vertices of C and let i; be a vertex of G. The representation γ(v; W) of v with respect to W is the κ-tuple {d{v1,w1),d{v,w 2), d{v,Wk)). 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 cardi-nality is the metric dimension of G, denoted by dim(G). A family Q of connected graphs is a family with constant metric dimension if dim{G) does not depend upon the choice of G in Q. In this paper, we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladder M constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder Mn except when n = 2(mod 8). It is natural to ask for the charac-terization of regular graphs with constant metric dimension.
Original language | English |
---|---|
Pages (from-to) | 403-410 |
Number of pages | 8 |
Journal | Ars Combinatoria |
Volume | 105 |
Publication status | Published - Jul 2012 |
Externally published | Yes |
Keywords
- Basis
- Metric dimension
- Möbius ladder
- Resolving set
ASJC Scopus subject areas
- General Mathematics