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.
The author would like to thank the referees for careful reading and helpful comments that improved the readability of the paper. The work was supported by the Basic Science Research Program through the National Research Foundation of Korea ( NRF-2022R1F1A1069500 ). This research was done when the author was visiting University of Victoria for her sabbatical year.
