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

一组小号在图中的顶点的ģ是一个控制集如果每个顶点不是在小号相邻于一个顶点 小号。此外，如果S是一个独立集合，则S是一个独立的支配集合。独立控制数我（ģ的）ģ是在一个独立的控制集的最小基数ģ。在Goddard和Henning（离散数学313：839–854，2013）中，我们推测，如果G是阶n的连通三次方图 ，则\（i（G）\ le \ frac {3} {8} n \）除外如果G是完整的二部图\（K_ {3,3} \）或5棱镜\（C_5 \，\ Box \，K_2 \）。此外，他们构造了两个无限的连通立方图族，其独立支配度为八分之三。在本文中，我们提供了一个新的n阶连通三次图族G，使得\（i（G）= \ frac {3} {8} n \）。我们还表明，如果G是阶n的次三次图，没有孤立的顶点，则\（i（G）\ le \ frac {1} {2} n \），并且我们表征在该范围内实现相等的图。