On the multipacking number of grid graphs

TL;DR

This paper proves that for large grid graphs with height at least 4 and width at least 7, the maximum multipacking number equals the broadcast domination number, extending known results beyond chordal graphs.

Contribution

It establishes the equality of broadcast and multipacking numbers for large grid graphs, which are not chordal, broadening the class of graphs where this equality holds.

Findings

mp(G) = γ_b(G) for large grid graphs with height ≥ 4 and width ≥ 7

Extends known equality from strongly chordal graphs to certain grid graphs

Provides insights into the structure of multipacking and broadcast domination in non-chordal graphs

Abstract

In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph $G$ is denoted $\gamma_b(G)$. The dual of this problem is called multipacking: a multipacking is a set $M$ of vertices such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$ . The maximum size of a multipacking in a graph $G$ is denoted mp(G). Naturally mp(G) $\leq \gamma_b(G)$. Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal.