# Relation Between Broadcast Domination and Multipacking Numbers on Chordal Graphs

## Document Type

Conference Article

## Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

## Abstract

For a graph G= (V, E) with a vertex set V and an edge set E, a function f: V→ { 0, 1, 2,.., diam(G) } is called a broadcast on G. For each vertex u∈ V, if there exists a vertex v in G (possibly, u= v ) such that f(v) > 0 and d(u, v) ≤ f(v), then f is called a dominating broadcast on G. The cost of the dominating broadcast f is the quantity ∑ v∈Vf(v). The minimum cost of a dominating broadcast is the broadcast domination number of G, denoted by γb(G). A multipacking is a set S⊆ V in a graph G= (V, E) such that for every vertex v∈ V and for every integer r≥ 1, the ball of radius r around v contains at most r vertices of S, that is, there are at most r vertices in S at a distance at most r from v in G. The multipacking number of G is the maximum cardinality of a multipacking of G and is denoted by mp(G). It is known that mp(G)≤γb(G) and that γb(G)≤2mp(G)+3 for any graph G, and it was shown that γb(G)-mp(G) can be arbitrarily large for connected graphs (as there exist infinitely many connected graphs G where γb(G)/mp(G)=4/3 with mp(G) arbitrarily large). For strongly chordal graphs, it is known that mp(G)=γb(G) always holds. We show that, for any connected chordal graph G, γb(G)≤⌈32mp(G)⌉. We also show that γb(G)-mp(G) can be arbitrarily large for connected chordal graphs by constructing an infinite family of connected chordal graphs such that the ratio γb(G)/mp(G)=10/9, with mp(G) arbitrarily large. This result shows that, for chordal graphs, we cannot improve the bound γb(G)≤⌈32mp(G)⌉ to a bound in the form γb(G)≤c1·mp(G)+c2, for any constant c1< 10 / 9 and c2.

## First Page

297

## Last Page

308

## DOI

10.1007/978-3-031-25211-2_23

## Publication Date

1-1-2023

## Recommended Citation

Das, Sandip; Foucaud, Florent; Islam, Sk Samim; and Mukherjee, Joydeep, "Relation Between Broadcast Domination and Multipacking Numbers on Chordal Graphs" (2023). *Conference Articles*. 618.

https://digitalcommons.isical.ac.in/conf-articles/618