Broadcast Dimension of Graphs
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In this paper we initiate the study of broadcast dimension, a variant of metric dimension. Let G be a graph with vertex set V(G), and let d(u,w) denote the length of a u-w geodesic in G. For k ≥ 1, let d_k(x,y)=min{d(x,y), k+1}. A function f: V(G) →ℤ^+ ∪{0} is called a resolving broadcast of G if, for any distinct x,y ∈ V(G), there exists a vertex z ∈ V(G) such that f(z)=i>0 and d_i(x,z) ≠ d_i(y,z). The broadcast dimension, bdim(G), of G is the minimum of c_f(G)=∑_v ∈ V(G) f(v) over all resolving broadcasts of G, where c_f(G) can be viewed as the total cost of the transmitters (of various strength) used in resolving the entire network described by the graph G. Note that bdim(G) reduces to adim(G) (the adjacency dimension of G, introduced by Jannesari and Omoomi in 2012) if the codomain of resolving broadcasts is restricted to {0,1}. We determine its value for cycles, paths, and other families of graphs. We prove that bdim(G) = Ω(logn) for all graphs G of order n, and that the result is sharp up to a constant factor. We show that adim(G)/bdim(G) and bdim(G)/dim(G) can both be arbitrarily large, where dim(G) denotes the metric dimension of G. We also examine the effect of vertex deletion on the adjacency dimension and the broadcast dimension of graphs.
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