Independent domination in subcubic graphs

A set S of vertices in a graph G is dominating if every vertex not adjacent to S. If, addition, an independent set, then set. The domination number i(G) the minimum cardinality G. In Goddard and Henning (Discrete Math 313:839–854, 2013) conjectured that connected cubic order n, $$i(G) \le \frac{3}{8}n$$ , except complete bipartite $$K_{3,3}$$ or 5-prism $$C_5 \, \Box K_2$$ . Further they construct two infinite families graphs with three-eighths their order. this paper, we provide new family n such = We also show subcubic no isolated vertex, \frac{1}{2}n$$ characterize achieving equality bound.