2-limited broadcast domination in subcubic graphs

In this paper, we consider a limited version of the well-studied dominating broadcast function of a graph. For a graph G, a function f:V(G)→{0,1,2} is a 2-limited dominating broadcast if for each vertex u in G there exist a vertex v such that f(v)>0 and d_G(u,v)≤f(v), where d_G(u,v) denotes the distance between u and v in G. The cost of a 2-limited dominating broadcast f is the sum of the function values f(v) summed over all vertices v in G. The 2-limited broadcast domination number of G, denoted by γ_b,2(G), is the minimum cost of a 2-limited dominating broadcast in G. We observe that γ_b,2(G)≤γ(G), where γ(G) is the domination number of G. A graph is (C₄,C₆)-free if it does not contain a 4-cycle or a 6-cycle as an induced subgraph. We prove that if G is a cubic graph of order n that is (C₄,C₆)-free, then [Formula presented].