Abstract
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).
Similar content being viewed by others
References
Bouchemakh, I., Boumali, A.: Broadcast domination number of the cross product of paths. In: ODSA 2010 Conference, Universität Rostock, September 13–15 (2010)
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)
Cockayne, E.J., Herke, S., Mynhardt, C.M.: Broadcasts and domination in trees. Discrete Math. (2009). doi:10.1016/j.disc.2009.12.012
Dabney J., Dean B.C., Hedetniemi S.T.: A linear-time algorithm for broadcast domination in a tree. Networks 53(2), 160–169 (2009)
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)
Erwin, D.: Cost domination in graphs. Dissertation, Western Michigan University (2001)
Erwin D.: Dominating broadcasts in graphs. Bull. Inst. Combin. Appl. 42, 89–105 (2004)
Heggernes P., Lokshtanov D.: Optimal broadcast domination in polynomial time. Discrete Math. 36, 3267–3280 (2006)
Herke S., Mynhardt C.M.: Radial trees. Discrete Math. 309, 5950–5962 (2009)
Seager, S.M.: Dominating Broadcasts of Caterpillars, Ars Comb. 88 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bouchemakh, I., Zemir, M. On the Broadcast Independence Number of Grid Graph. Graphs and Combinatorics 30, 83–100 (2014). https://doi.org/10.1007/s00373-012-1253-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-012-1253-0