## Faculty Publications & Research

## Document Type

Article

## Publication Date

2013

## Disciplines

Mathematics

## Abstract

Let G be a weighted graph in which each vertex initially has weight 1. A *total acquisition move* transfers all the weight from a vertex *u* to a neighboring vertex *v*, under the condition that before the move the weight on *v* is at least as large as the weight on *u*. The *(total)* *acquisition number* of *G*, written *a _{t}*(

*G*), is the minimum size of the set of vertices with positive weight after a sequence of total acquisition moves. Among connected

*n*-vertex graphs,

*a*(

_{t}*G*) is maximized by trees. The maximum is Θ(√(

*n*lg

*n*) for trees with diameter 4 or 5. It is⌊(

*n*+ 1)/3⌋ for trees with diameter between 6 and (2/3)(

*n*+ 1), and it is⌈(2

*n*– 1 –

*D*)/4⌉ for trees with diameter

*D*when (2/3)(

*n*+ 1) ≤

*D*≤

*n*- 1. We characterize trees with acquisition number 1, which permits testing

*a*(

_{t}*G*) ≤

*k*in time

*O*(

*n*

^{k}^{+2}) on trees. If

*G ≠ C*, then min{

_{5}*a*(

_{t}*G*),

*a*()} = 1. If

_{t}*G*has diameter 2, then

*a*(

_{t}*G*) ≤ 32 ln

*n*ln ln

*n*; we conjecture a constant upper bound. Indeed,

*a*(

_{t}*G*) = 1 when

*G*has diameter 2 and no 4-cycle, except for four graphs with acquisition number 2. Deleting one edge of an

*n*-vertex graph cannot increase

*a*by more than 6.84√

_{t}*n*, but we construct an

*n*-vertex tree with an edge whose deletion increases it by more than (1/2)√

*n*. We also obtain multiplicative upper bounds under products.

## Recommended Citation

Lesaulnier, T.D., Prince, N., Wenger, P.S., West, D.B., & Worah, P. (2013). Total Acquistion in Graphs. SIAM Journal of Discrete Mathematics, 27(4), 1800-1819.