The Fractional Local Metric Dimension of Graphs
DOI:
https://doi.org/10.55016/ojs/cdm.v19i3.62807Keywords:
Local metric dimension, Fractional local metric dimension, Strong product of graphs, Cartesian product of graphsAbstract
The fractional versions of graph-theoretic invariants multiply the range of applications in scheduling, assignment and operational research problems. For this interesting aspect of fractional graph theory, we introduce the fractional version of local metric dimension of graphs. The local resolving neighborhood $L(xy)$ of an edge $xy$ of a graph $G$ is the set of those vertices in $G$ which resolve the vertices $x$ and $y$. A function $f:V(G)\rightarrow[0, 1]$ is a local resolving function of $G$ if $f(L(xy))\geq1$ for all edges $xy$ in $G$. The minimum value of $f(V(G))$ among all local resolving functions $f$ of $G$ is the fractional local metric dimension of $G$. We study the properties and bounds of fractional local metric dimension of graphs and give some characterization results. We determine the fractional local metric dimension of strong and Cartesian product of graphs.
References
of graphs, Disc. Math., 312(2012), 1584-1590.
\bibitem{AM2}
S. Arumugam, V. Mathew, J. Shen, On fractional metric dimension
of graphs, Disc. Math. Algorithms and Appl., (2013).
\bibitem{2}
G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann, Resolvability in
graphs and the metric dimension of a graph, Disc. Appl. Math.,
32(2000), 105, 99-113.
\bibitem{GCPZ}
G. Chartrand, C. Poisson, P. Zhang, Resolvability and the upper
dimension of graphs, Comput. Math. Appl., 39 (2000), 19-28.
\bibitem{1}
G. Chartrand, L. Lesniak, Graphs and Digraphs, $3$rd ed., Chapman
and Hall, London, 1996.
\bibitem{3}
V. Chvatal, Mastermind Combinatorica, 1983, 3, 325-329.
\bibitem{CO}
J. Currie, O. R. Ollermann, The metric dimension and
metric independence of a graph, J. Combin. Math. Combin. Comput.,
39(2001), 157-167.
\bibitem{feh}
M. Fehr, S. Gosselin, O. R. Oellermann, The metric dimension of
Cayley digraphs, Disc. Math., 306(2006), 31-41.
\bibitem{FLW}
M. Feng, B. Lv, K. Wang, On the fractional metric dimension of
graphs, Disc. Appl. Math., 170(2014), 55-63.
\bibitem{FW}
M. Feng, K. Wang, On the fractional metric dimension of corona product graphs and lexicographic product
graphs, arXiv:1206.1906v1 [math.CO].
\bibitem{FW2}
M. Feng, K. Wang, On the metric dimension and fractional metric dimension of hierarchical product of
graphs, Appl. Anal. Disc. Math., 7(2013), 302-313.
\bibitem {har}
F. Harary, R. A. Melter, On the metric dimension of a graph, Ars
Comb., 2(1976), 191-195.
\bibitem{11}
R. Hammack, W. Imrich, S. Klav\v{z}ar, Handbook of product graphs,
Discrete Mathematics and its Applications, 2nd ed., CRC Press, 2011.
URL http://www.crcpress.com/product/isbn/9781439813041
\bibitem{KS}
D. A. Krismanto, S. W. Saputro, Fractional metric dimension of tree
and unicyclic Graph, Procedia Comput. Sci., 74(2015), 47-52.
\bibitem{7}
S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs,
Disc. Appl. Math., 70(1996), 217-229.
\bibitem{10}
F. Okamoto, B. Phinezy, P. Zhang, The local metric dimension of a
graph, Math. Bohem., 135(2010), 3, 239-255.
\bibitem{8}
H. Shapiro, S. Soderberg, A combinatory detection problem, Amer.
Math. Monthly, 70(1963), 1066-1070.
\bibitem{9}
PJ. Slater, Leaves of trees, Congr. Numer., 14(1975), 549-559.
\bibitem{EY}
E. Yi, The fractional metric dimension of permutation
graphs, Acta Math. Sin. (Engl. Ser.), 31(2015), 367-382.
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