Abstract A set S of vertices is an independent dominating set if it is both independent and dominating, and the idomatic number is the maximum number of vertex-disjoint independent dominating sets. In this paper we consider a fractional version of this. Namely, we define the fractional idomatic number as the maximum ratio | F | / m ( F ) over all families F of independent dominating sets, where m ( F ) denotes the maximum number of times an element appears in F . We start with some bounds including a connection with dynamic colorings. Then we show that the independent domination number of a planar graph with minimum degree 2 is at most half its order, and its fractional idomatic number is at least 2. Moreover, we show that an outerplanar graph of minimum degree 2 has idomatic number at least 2. We conclude by providing formulas for the parameters for the join, disjoint union and lexicographic product of graphs, while providing some bounds for cubic graphs.

中文翻译：

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图的独立支配、着色和分数惯用数

摘要 顶点集合S如果既是独立又是支配集，则是一个独立支配集，惯用数是顶点不相交的独立支配集的最大数目。在本文中，我们考虑了它的一个部分版本。即，我们将分数惯用数定义为最大比率 | F | / m ( F ) 在独立支配集的所有族 F 上，其中 m ( F ) 表示元素在 F 中出现的最大次数。我们从一些边界开始，包括与动态着色的联系。然后证明最小度为 2 的平面图的独立支配数至多是它的阶数的一半，并且它的分数依恋数至少是 2。此外，我们证明了一个最小度为 2 的外平面图至少有一个依恋数2.