Graphs and Combinatorics

, Volume 30, Issue 1, pp 83–100 | Cite as

On the Broadcast Independence Number of Grid Graph

Original Paper


A broadcast on a nontrivial connected graph G is a function \({f:V \longrightarrow \{0, \ldots,\operatorname{diam}(G)\}}\) such that for every vertex \({v \in V(G)}\) , \({f(v) \leq e(v)}\) , where \({\operatorname{diam}(G)}\) denotes the diameter of G and e(v) denotes the eccentricity of vertex v. The broadcast independence number is the maximum value of \({\sum_{v \in V} f(v)}\) over all broadcasts f that satisfy \({d(u,v) > \max \{f(u), f(v)\}}\) for every pair of distinct vertices u, v with positive values. We determine this invariant for grid graphs \({G_{m,n} = P_m \square P_n}\) , where \({2 \leq m \leq n}\) and □ denotes the Cartesian product. We hereby answer one of the open problems raised by Dunbar et al. in (Discrete Appl Math 154:59–75, 2006).


Grid graph Broadcast Independent set 


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  1. 1.
    Bouchemakh, I., Boumali, A.: Broadcast domination number of the cross product of paths. In: ODSA 2010 Conference, Universität Rostock, September 13–15 (2010)Google Scholar
  2. 2.
    Blair J.R.S., Heggernes P., Horton S.B., Maine F.: Broadcast domination algorithms for interval graphs, series–parallel graphs and trees. Congr. Num. 169, 55–77 (2004)MATHGoogle Scholar
  3. 3.
    Cockayne, E.J., Herke, S., Mynhardt, C.M.: Broadcasts and domination in trees. Discrete Math. (2009). doi: 10.1016/j.disc.2009.12.012
  4. 4.
    Dabney J., Dean B.C., Hedetniemi S.T.: A linear-time algorithm for broadcast domination in a tree. Networks 53(2), 160–169 (2009)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Dunbar J.E., Erwin D.J., Haynes T.W., Hedetniemi S.M., Hedetniemi S.T.: Broadcasts in graphs. Discrete Appl. Math. 154, 59–75 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Erwin, D.: Cost domination in graphs. Dissertation, Western Michigan University (2001)Google Scholar
  7. 7.
    Erwin D.: Dominating broadcasts in graphs. Bull. Inst. Combin. Appl. 42, 89–105 (2004)MATHMathSciNetGoogle Scholar
  8. 8.
    Heggernes P., Lokshtanov D.: Optimal broadcast domination in polynomial time. Discrete Math. 36, 3267–3280 (2006)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Herke S., Mynhardt C.M.: Radial trees. Discrete Math. 309, 5950–5962 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Seager, S.M.: Dominating Broadcasts of Caterpillars, Ars Comb. 88 (2008)Google Scholar

Copyright information

© Springer Japan 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics, Laboratory LAID3University of Sciences and Technology Houari Boumediene(USTHB)AlgiersAlgeria
  2. 2.Department of MathematicsUniversity of BlidaBlidaAlgeria

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