Broadcast Domination in Line Graphs of Trees

Abstract

A dominating broadcast of a graph G is a function f : V ( G ) → { 0 , 1 , 2 , … , diam ( G ) } such that f ( v ) ⩽ e ( v ) for all v ∈ V ( G ) , where e ( v ) is the eccentricity of v , and for every vertex u ∈ V ( G ) , there exists a vertex v with f ( v ) > 0 and d ( u , v ) ⩽ f ( v ) . The cost of f is ∑ v ∈ V ( G ) f ( v ) . The minimum of costs over all the dominating broadcasts of G is called the broadcast domination number γ b ( G ) of G . A graph $G$ is said to be radial if γ b ( G ) = rad ( G ) . In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph L ( G ) of G , in terms of γ b ( G ) , and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees T for which L ( T ) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the i -subdivision graph of K 1 , n and a subclass of caterpillars are radial. Also, we show that γ b ( L ( C ) ) = γ ( L ( C ) ) for any caterpillar C .