Locating-Total Domination in Grid Graphs

  • Jia GuoEmail author
  • Zhuo Li
  • Mei Lu


Let \(G=(V,E)\) be a graph with no isolated vertex. A subset \(S\subseteq V(G)\) is a total dominating set of graph G if every vertex in V(G) is adjacent to at least one vertex in S. A total dominating set S of graph G is a locating-total dominating set if for every pair of distinct vertices \(u_1\) and \(u_2\) in \(V(G)-S\), \(N(u_1)\cap S\ne N(u_2)\cap S\). The locating-total domination number of graph G, denoted by \(\gamma _t^L(G)\), is the minimum cardinality of a locating-total dominating set of G. In this paper, we investigate the bounds of locating-total domination number of grid graphs.


Locating-total dominating set Locating-total domination number Cartesian product Grid graph 

Mathematics Subject Classification

05C50 15A18 



This work is partially supported by National Natural Science Foundation of China (Nos. 11801450, 11771247). In addition, the authors are thankful to the anonymous referees for their useful comments and suggestions.


  1. 1.
    Blidia, M., Chellali, M., Maffray, F., Moncel, J., Semri, A.: Locating-dominaiton and identifying codes in trees. Australas. J. Comb. 39, 219–232 (2007)zbMATHGoogle Scholar
  2. 2.
    Blidia, M., Dali, W.: A characterization of locating-total domination edge critical graphs. Discuss. Math. Graph Theory 31(1), 197–202 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chellali, M.: On locating and differentiating-total domination in trees. Discuss. Math. Graph Theory 28(3), 383–392 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chellali, M., Jafari Rad, N.: Locating-total domination critical graphs. Australas. J. Comb. 45, 227–234 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, X.G., Sohn, M.Y.: Bounds on the locating-total domination number of a tree. Discrete Appl. Math. 159, 769–773 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Haynes, T.W., Henning, M.A., Howard, J.: Locating and total dominating sets in trees. Discrete Appl. Math. 154(8), 1293–1300 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Henning, M.A., Rad, N.J.: Locating-total domination in graphs. Discrete Appl. Math. 160, 1986–1993 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Miller, M., Rajan, R.S., Jayagopal, R., Rajasingh, I., Manuel, P.: A note on the locating-total domination in graphs. Discuss. Math. Graph Theory 37(3), 745–754 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ning, W., Lu, M., Wang, K.: Bounding the locating-total domination number of a tree in terms of its annihilation number. Discuss. Math. Graph Theory 39(1), 31–40 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rad, N.J., Rahbani, H.: A note on the locating-total domination number in trees. Australas. J. Comb. 66(3), 420–424 (2016)zbMATHGoogle Scholar
  11. 11.
    Xing, H., Sohn, M.Y.: Bounds on locating total domination number of the Cartesian product of cycles and paths. Inf. Process. Lett. 115(12), 950–956 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.College of ScienceNorthwest A&F UniversityYanglingPeople’s Republic of China
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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