Total Acquisition on Grids
Lori MacDonald, Paul S. Wenger, Scott Wright · Australas. J Comb. · 2013
On a weighted graph G, a total acquisition move transfers weight from a vertex u to a neighbor v if the weight on v is at least as much as the weight on u. Starting with all vertices having weight 1, the total acquisition number of G, denoted at(G), is the minimum number of vertices with positive weight after a sequence of total acquisition moves. In [D. Lampert and P. Slater, The acquisition number of a graph, Congr. Numer. 109 (1995), 203–210] it is shown that at(G) � � jV (G)j/2 �(G) � for all G, and P52P5 is given as an example where this bound is not sharp. In this paper, we determine at(Pn2Pm) exactly when n and m are not 5 and give nontrivial upper and lower bounds on at(Pn2P5).