Total Acquisition in Graphs

Authors

Timothy D. Lesaulnier
Noah Prince, Illinois Math and Science AcademyFollow
Paul S. Wenger
Douglas B. West
Pratik Worah

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⌈(2n – 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 ≠ C₅, then min{a_t(G), a_t()} = 1. If 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_t by more than 6.84√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.