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.

297

308

DOI

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

1-1-2023

COinS