On the broadcast independence number of locally uniform 2-lobsters

Ahmane, Messaouda ; Bouchemakh, Isma  ; Sopena, Éric

Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 199-230

Résumé

Let G be a simple undirected graph. A broadcast on G is a function f : V(G)→ℕ such that f(v)≤ 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)=∑_v∈ V(G)f(v). A broadcast f on G is independent if for every two distinct vertices u and v in G with f(u) gt;0 and f(v) gt;0, d_G(u,v) gt;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 Appl. Math. 244 (2018) 20–35], 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.

Détail

Keywords: independence, broadcast independence, lobster