On the Broadcast Independence Number of Caterpillars
Résumé
Let $G$ be a simple undirected graph.
A broadcast on $G$ is
a function $f : V(G)\rightarrow\mathbb{N}$ such that $f(v)\le e_G(v)$ holds for every vertex $v$ of $G$,
where $e_G(v)$ denotes the eccentricity of $v$ in $G$, that is, the maximum distance from $v$ to any other vertex of $G$.
The cost of $f$ is the value ${\rm cost}(f)=\sum_{v\in V(G)}f(v)$.
A broadcast $f$ on $G$ is independent if for every two distinct vertices $u$ and $v$ in $G$, $d_G(u,v)>\max\{f(u),f(v)\}$,
where $d_G(u,v)$ denotes the distance between $u$ and $v$ in $G$.
The broadcast independence number of $G$ is then defined as the maximum cost of an independent broadcast on $G$.
In this paper, we study independent broadcasts of caterpillars and give an explicit formula for the
broadcast independence number of caterpillars having no pair of adjacent trunks,
a trunk being an internal spine vertex with degree~2.
Domaines
Mathématique discrète [cs.DM]
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REV - ABS - On the Broadcast Independence Number of Caterpillars.pdf (374.35 Ko)
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