Journal of Advances in Mathematics and Computer Science

Bi-Metric Dimension of Graphs

A. Raghavendra

Department of Mathematics, Poornapajna College, Udupi, Karnataka State, India.

B. Sooryanarayana *

Department of Mathematics, Dr. Ambedkar Institute of Technology, Bangalore 560 056, Karnataka State, India.

Chandru Hegde

Department of Mathematics, Mangalore University, Konaje Mangalore 574 199, Karnataka State, India.

*Author to whom correspondence should be addressed.

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Abstract

For a connected graph G, a subset S = {s₁, s₂,...., s_k} of vertices of G and each vertex x of G we associate a pair of k-dimensional vectors (u,v), where u = (d(x, s₁), d(x, s₂),...., d(x, s_k)) and v = (δ(x, s₁), δ(x, s₂),...., (x, s_k)), where d(x, s_i) and δ(x, s_i) respectively denote the lengths of a shortest and longest paths between x and s_i. The subset S is said to bi-resolve G if no two distinct vertices receive the same pair. The minimum cardinality of a bi-resolving set is called bi-metric dimension of G. In this paper we show bi-metric dimension is lesser than or equal to the metric dimension and determine bi-metric dimensions of some standard graphs.

Keywords: Metric Dimension, Landmarks, Bi-Metric dimension.

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