Multipackings in Graphs

Abstract

A vertex subset M of a graph G is a multipacking if for each vertex v, and each positive integer s less than or equal to the diameter of G, v is within distance s of at most s vertices of M. The multipacking number of a graph is the maximum cardinality of a multipacking of G. A generalization of 2-packings, multipackings offer interesting insight into the minimum cost broadcast domination problem. This paper surveys recent results in the study of multipackings, including the equality of the multipacking number and broadcast number in trees, an extension of Farber's Algorithm for finding dominating sets and 2-packings of strongly chordal graphs, and some early results on fractional multipackings. The paper closes with a series of in-depth examples of the various algorithms presented.