Let G = (V, E) be a connected graph. A dominator sequence in G is a sequence of vertices S = (v1, v2,⋯, vk) such that for each i with 2 ≤ i ≤ k, the vertex vi dominates at least one vertex which is not dominated by v1, v2, ⋯, vi-1. If further the set of vertices in S is an independent set, then S is called an independent dominator sequence (IDS) in G. The maximum length of an IDS in G is called the independent dominator sequence number of G and is denoted by lι(G). In this paper we initiate a study of this parameter. © 2015 The Authors.