Graphs and Combinatorics

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

On the Broadcast Independence Number of Grid Graph

Original Paper

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).

Keywords

Grid graph Broadcast Independent set 

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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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