A set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent 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. The independent domination subdivision number \( \hbox {sd}_{\mathrm{i}}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter \( \hbox {sd}_{\mathrm{i}}(G)\) is well defined and differs significantly from the well-studied domination subdivision number \(\mathrm{sd_\gamma }(G)\). For example, if G is a block graph, then \(\mathrm{sd_\gamma }(G) \le 3\), while \( \hbox {sd}_{\mathrm{i}}(G)\) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree \(\Delta (G)\) such that \( \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2\), in contrast to the known result that \(\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1\) always holds. Among other results, we present a simple characterization of trees T with \( \hbox {sd}_{\mathrm{i}}(T) = 1\).

一组小号在图的顶点的摹是控制集如果每个顶点不在小号是广告jacent到顶点 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。独立的控制细分数\（\ hbox {sd} _ {\ mathrm {i}}（G）\）是必须细分的最小边数（G中的每个边（最多可以细分一次），以增加独立的统治人数。我们表明，对于至少三个顶点上的每个连通图G，参数\（\ hbox {sd} _ {\ mathrm {i}}（G）\）定义明确，并且与经过充分研究的支配细分数显着不同\（\ mathrm {sd_ \ gamma}（G）\）。例如，如果G是程序框图，则\（\ mathrm {sd_ \ gamma}（G）\ le 3 \），而\（\ hbox {sd} _ {\ mathrm {i}}（G）\）可以任意大。进一步，我们表明存在具有任意大的最大度\（\ Delta（G）\）的连通图G， 使得\（\ hbox {sd} _ {\ mathrm {i}}（G）\ ge 3 \ Delta（G）-2 \），与已知结果\（\ mathrm {sd_ \ gamma}（G）相反\ le 2 \ Delta（G）-1 \）始终成立。在其他结果中，我们用\（\ hbox {sd} _ {\ mathrm {i}}（T）= 1 \）来表示树T的简单特征。