Weighted total acquisition

In the Weighted Total Acquisition problem (WTA) on a weighted graph G (only positive non-zero weights), vertex v can acquire total weight of neighbour u if and only current is at least that u. The new then zero, sum weights just before acquisition. Over all possible acquisition sequences in with function w, minimum number vertices end denoted by at(G,w). Given G, weighting an integer k?1, WTA asks whether at(G,w)?k. Unary (Binary resp.) corresponds to when are encoded unary (binary resp.). We prove polynomial-time solvable graphs bounded treewidth degree. When bounded, this algorithm quasi-polynomial, i.e., it runs time WO(logW), where W vertices. Moreover, we show FPT trees parameterized maximum On negative side, NP-complete trivially perfect split graphs, even k=1 latter. Binary degree, depth, wheels, but XP for wheels solution size. K3,n, planar pathwidth 2, unit interval k=1, k?2 (but k=1).