Metric dimension of a direct sum and direct product of metric spaces

Abstract

For an arbitrary metric space (X, d) subset A \subset X is called resolving if for any two points x \ne y \in X there exists point a in subset A for which following inequality holds d(a, x) \ne d(a, y). Cardinality of the subset A with the least amount of points is called metric dimension.

In general, the problem of finding metric dimension of a metric space is NP–hard [1]. In this paper metric dimension for particular constructs of metric spaces is provided. In particular, it is fully characterized metric dimension for the direct sum of metric spaces and shown some properties of the metric dimension of direct product.

Key words: metric dimension, direct sum of metric spaces, direct product of metric spaces.

Pages of the article in the issue: 41 - 46

Language of the article: Ukrainian

References

GAREY, M. R. and JOHNSON, D. S. (1990) “Computers and Intractability; A Guide to the Theory of NP-Completeness”, W. H. Freeman & Co., New York.

BLUMENTHAL, L. M. (1953) “Theory and applications of distance geometry”, Clarendon Press, Oxford.

HARARY, F. and MELTER R. A. (1976) “On the Metric Dimension of a Graph” Ars Combin. 20, pp. 191–195.

SLATER, P. J. (1975) “Leaves of Trees”, Congr. Numer., 14, pp. 549–559.

SEBO, A. and TANNIER, E. (2004) “On Metric Generators of Graphs”, Math. Oper. Res., 29 (2), pp. 383–393.

DUDENKO, M. and OLIYNYK, B. (2018) “On unicyclic graphs of metric dimension 2 with vertices of degree 4”, Algebra Discrete Math., 26 (2), pp. 256–269.

RODRIGUEZ-VELAZQUEZ, J. A. and YERO, I. G. (2010) “A note on the partition dimension of Cartesian product graphs”, Appl. Math. Comput., 217, pp. 3571–3574.

BAU, S. and BEARDON, A. (2013) “The Metric Dimension of Metric Spaces”, Comput. Methods Funct. Theory, 13.

HEYDARPOUR, M. and SAEID, M. (2014) “The metric dimension of geometric spaces”, Topology Appl., 178, pp. 230–235.

OLIYNYK, B. (2008) “k-homogeneity of the n-fold direct sum of metric space over itself”, Bull Univ. Kyiv, pp. 3639.

SEARCOID, M. (2007) “Metrics”, Springer, London.

How to Cite
Ponomarchuk, B. S. (1). Metric dimension of a direct sum and direct product of metric spaces. Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, (1-2), 41-46. https://doi.org/10.17721/1812-5409.2020/1-2.6
Section
Algebra, Geometry and Probability Theory