The partition dimension of a subdivision of a homogeneous firecracker
Abstract
Finding the partition dimension of a graph is one of the interesting (and uncompletely solved) problems of graph theory. For instance, the values of the partition dimensions for most kind of trees are still unknown. Although for several classes of trees such as paths, stars, caterpillars, homogeneous firecrackers and others, we do know their partition dimensions. In this paper, we determine the partition dimension of a subdivision of a particular tree, namely homogeneous firecrackers. Let G be any graph. For any positive integer k and e \in E(G), a subdivision of a graph G, denoted by S(G(e;k)), is the graph obtained from G by replacing an edge $e$ with a (k+1)-path. We show that the partition dimension of S(G(e;k)) is equal to the partition dimension of G if G is a homogeneous firecracker. We show that the partition dimension of S(G(e;k)) is equal to the partition dimension of G if G is a homogeneous firecracker.
Keywords
Full Text:
PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.2.20
References
S. Akhter and R. Farooq. (2019). Metric dimension of fullerene graphs, Electron. J. Graph Theory Appl. 7 (1) (2019), 91--103.
Amrullah, H. Assiyatun, E.T. Baskoro, S. Uttunggadewa, R. Simanjuntak, The partition dimension for a subdivision of homogeneous caterpillars, AKCE Int. J. Graphs Comb. 10 (3) (2013), 309--315.
Amrullah, S. Azmi, H. Soeprianto, M. Turmuzi, and Y.S. Anwar,
The partition dimension of subdivision graph on the star, J. Phys. Conf. Ser. 1280 (2) (2019), 022037.
Amrullah, E.T. Baskoro, S. Rinovia and S. Uttunggadewa. The partition dimension of a subdivision of a complete graph, Procedia Comput. Sci. 74 (2015), 53--59.
Amrullah, E.T. Baskoro, S. Uttunggadewa, and R. Simanjuntak. The partition dimension of subdivision of a graph. AIP Conf. Proc. 1707 (2016), (1) 020001.
Amrullah, Darmaji and E.T. Baskoro, The partition dimension for homogeneous firecrackers, Far East J. Appl. Math. 90 (2015), 77--98.
E.T. Baskoro and Darmaji, The partition dimension of the corona product of two graphs, Far East J. Math. Sci. 66 (2012), 181--196.
E.T. Baskoro and D.O. Haryeni, All graphs of order n >= 11 and diameter 2 with partition dimension n-3, Heliyon 6 (2020) e03694.
G. Chartrand, E. Salehi, and P. Zhang, On the partition dimension of graph, Congr. Numer. 131 (1998), 55--66.
G. Chartrand, E. Salehi, and P. Zhang, The partition dimension of graph, Aequationes Math. 59 (2000), 45--54.
G. Chartrand P. Zhang, The theory and applications of resolvability in graphs: a survey, Congr. Numer. 160 (2003), 47--68.
Darmaji, S. Uttunggadewa, R. Simanjuntak, E.T. Baskoro, The partition dimension of a complete multipartite graph, a special caterpillar and a windmill, J. Comb. Math. Comb. Comput. 71 (2009), 209--215.
F. Harary and R. Melter, On the metric dimension of a graph, Ars Comb. 2 (1976), 191--195.
D.O. Haryeni, E.T. Baskoro, and S.W. Saputro. A method to construct graphs with certain partition dimension, Electron. J. Graph Theory Appl. 7 (2), (2019), 251--263.
R.A. Juan, I.G. Yero, and M. Lemanska. On the partition dimension of trees. Discrete Appl. Math. 166 (2014), 204--209.
M.A. Johnson, Structure-activity maps for visualizing the graph variables arising in drug design, J. Biopharm. Statist. 3 (1993), 203--236.
R.A. Melter, I. Tomescu, Metric bases in digital geometry, Comput. Vis., Graph. Im. Proc. 25 (1984), 113--121.
P. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549--559.
R.A. Solekhah and T.A. Kusmayadi, On the local metric dimension of t-fold wheel, Pn O Km, and generalized fan, Indonesian Journal of Combinatorics 2 (2) (2018), 88--96.
L. Yulianti, Narwen and S. Hariyani, On the subdivided thorn graph
and its metric dimension, Indonesian Journal of Combinatorics 3 (1) (2019), 34--40.
Refbacks
- There are currently no refbacks.
ISSN: 2338-2287
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.