Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

On the Metric Dimension of Corona Product of a Graph with K1

  • Mohsen Jannesari (Department of Science, Shahreza Campus, University of Isfahan)
  • Received : 2021.11.17
  • Accepted : 2022.06.24
  • Published : 2023.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the k-vector r(v|W) = (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, G ⊙ H of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(G ⊙ K1), characterize all graphs G with dim(G ⊙ K1) = dim(G), and prove that dim(G ⊙ K1) = n - 1 if and only if G is the complete graph Kn or the star graph K1,n-1.

Keywords

References

  1. Z. Beerliova, F. Eberhard, T. Erlebach, A. Hall, M. Hoffmann, M. Mihalak and L. S. Ram, Network dicovery and verification, IEEE J. Sel. Areas Commun., 24(12)(2006) 2168-2181. https://doi.org/10.1109/JSAC.2006.884015
  2. P. Buczkowski, G. Chartrand, C. Poisson and P. Zhang, On k-dimensional graphs and their bases, Period. Math. Hungar. 46(1)(2003), 9-15. https://doi.org/10.1023/A:1025745406160
  3. J. Caceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C. Seara and D. R. Wood, On the metric dimension of cartesian products of graphs, SIAM J. Discrete Math., 21(2)(2007) 423-441. https://doi.org/10.1137/050641867
  4. G. G. Chappell, J. Gimbel and C. Hartman, Bounds on the metric and partition dimensions of a graph, Ars Combin., 88(2008) 349-366.
  5. G. Chartrand, L. Eroh, M. A. Johnson and O. R. Ollermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105(2000) 99-113. https://doi.org/10.1016/S0166-218X(00)00198-0
  6. G. Chartrand and P. Zhang, The theory and applications of resolvability in graphs, Congr. Numer., 160(2003), 47-68.
  7. H. Fernau and J. A. Rodriguez-Velazquez, On the (adjacency) metric dimension of corona and strong product graphs and their local variants: Combinatorial and computational results, Discrete Appl. Math., 236(2018) 183-202. https://doi.org/10.1016/j.dam.2017.11.019
  8. F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin., 2(1976) 191-195.
  9. S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., 70(3)(1996) 217-229. https://doi.org/10.1016/0166-218X(95)00106-2
  10. R. A. Melter and I. Tomescu, Metric bases in digital geometry, Comput. Graph. Image Process., 25(1984) 113-121. https://doi.org/10.1016/0734-189X(84)90051-3
  11. A. Sebo and E. Tannier, On metric generators of graphs, Math. Oper. Res., 29(2)(2004) 383-393. https://doi.org/10.1287/moor.1030.0070
  12. P. J. Slater, Leaves of trees, Congr. Numer., 14(1975) 549-559.
  13. P. J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci., 22(1988) 445-455.
  14. I. G. Yero, D. Kuziak and J. A. Rodriguez-velaquez, On the metric dimension of corona product graphs, Comput. Math. Appl., 61(2011) 2793-2798.  https://doi.org/10.1016/j.camwa.2011.03.046