The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted I (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)sets are adjacent if and only if they differ by two adjacent vertices. Although not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H ∼ = I (G), all graphs have i-graphs. We determine the i-graphs of paths and cycles and, in the case of cycles, discuss the Hamiltonicity of these i-graphs.