Independent domination, colorings and the fractional idomatic number of a graph

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.