The Unit Acquisition Number of a Graph

TL;DR

This paper investigates the unit acquisition number of graphs, establishing bounds based on graph properties, characterizing cases of equality, and providing algorithms for specific graph classes.

Contribution

It introduces new bounds and exact values for the unit acquisition number, and develops efficient algorithms for computing it in certain graph families.

Findings

For graphs with minimum degree k, a_u(G) ≤ (n-1)/k, with equality for complete graphs and C5.

a_u(G) is at most the size of a maximum matching, with equality on some graphs.

Graphs with diameter 2 have a_u(G) = 1, except for C5 and Petersen graph.

Abstract

Let $G$ be a graph with nonnegative integer weights. A {\it unit acquisition move} transfers one unit of weight from a vertex to a neighbor that has at least as much weight. The {\it unit acquisition number} of a graph $G$, denoted $a_u(G)$, is the minimum size that the set of vertices with positive weight can be reduced to via successive unit acquisition moves when starting from the configuration in which every vertex has weight $1$. For a graph $G$ with $n$ vertices and minimum degree $k$, we prove $a_u(G)\le (n-1)/k$, with equality for complete graphs and $C_5$. Also $a_u(G)$ is at most the minimum size of a maximal matching in $G$, with equality on an infinite family of graphs. Furthermore, $a_u(G)$ is bounded by the maximum degree and by $\sqrt{n-1}$ when $G$ is an $n$-vertex tree with diameter at most $4$. We also construct arbitrarily large trees with maximum degree $5$ having unit acquisition number $1$, obtain a linear-time algorithm to compute the acquisition number of a caterpillar, and show that graphs with diameter $2$ have unit acquisition number $1$ except for $C_5$ and the Petersen graph.