Multipackings in Graphs

TL;DR

This paper surveys recent advances in the study of multipackings in graphs, exploring their properties, algorithms, and applications to broadcast domination problems.

Contribution

It compiles recent results on multipacking numbers, algorithms for special graph classes, and fractional multipacking concepts, providing a comprehensive overview.

Findings

Equality of multipacking and broadcast numbers in trees

Extension of Farber's Algorithm for certain graph classes

Initial results on fractional multipackings

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.