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 gamma(b, 2)(G), is the minimum cost of a 2-limited dominating broadcast in G. We observe that gamma(b, 2 )(G) <= gamma(G), where gamma(G) is the domination number of G. A graph is (C-4, C-6)-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 (C-4, C-6)-free, then gamma(b, 2 )(G) <= 1/3n. (C) 2020 Elsevier B.V. All rights reserved.