The Total Acquisition Number of Random Geometric Graphs

Let G be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex u to a neighbouring vertex v can be moved, provided that the weight on v is at least as large as the weight on u. The total acquisition number of G, denoted by at(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate random geometric graphs G(n, r) with n vertices distributed uniformly at random in [0, √ n]2 and two vertices being adjacent if and only if their distance is at most r. We show that asymptotically almost surely at(G(n, r)) = Θ(n/(r lg r)2) for the whole range of r = rn > 1 such that r lg r 6 √ n. By monotonicity, asymptotically almost surely at(G(n, r)) = Θ(n) if r < 1, and at(G(n, r)) = Θ(1) if r lg r > √ n. ∗The third author is supported in part by NSERC and Ryerson University the electronic journal of combinatorics 24(3) (2017), #P3.31 1