2-limited dominating broadcasts on cubic graphs without induced 4-cycles

Abstract

For a graph G, a function f:V(G)→{0,1,2} is called a 2-limited dominating broadcast on G if for every vertex u, there exists a vertex v such that f(v)>0 and the distance between u and v in G is at most f(v). The cost of f means the value ∑v∈V(G)f(v), and the 2-limited broadcast domination number γb,2(G) of G is the cost of a 2-limited dominating broadcast on G with minimum cost. Henning, MacGillivray and Yang (2020) conjectured that [Formula presented] for every cubic graph G, and then confirmed it for a cubic graph G having neither C4 nor C6 as an induced subgraph. In this paper, we improve their result, that is, we show that the conjecture holds for cubic graphs having no C4 as an induced subgraph.