On the broadcast independence number of locally uniform 2-lobsters

Abstract

Let~$G$ be a simple undirected graph. A broadcast on~$G$ is a function $f : V(G)\rightarrow\NNNNN$ 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 $\cost(f)=\sum_{v\in V(G)}f(v)$. 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~$G$. A caterpillar is a tree such that, after the removal of all leaf vertices, the remaining graph is a non-empty path. A lobster is a tree such that, after the removal of all leaf vertices, the remaining graph is a caterpillar. In [M. Ahmane, I. Bouchemakh and E. Sopena. On the Broadcast Independence Number of Caterpillars. Discrete Applied Mathematics, in press (2018)], we studied independent broadcasts of caterpillars. In this paper, carrying on with this line of research, we consider independent broadcasts of lobsters and give an explicit formula for the broadcast independence number of a family of lobsters called locally uniform $2$-lobsters.