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AbstractA set D of vertices in a connected graph G is said to be an independent 2-point set dominating set (or in short i-2psd set) of G if D is an independent set and for every non-empty subset [Formula: see text] there exists a non-empty subset [Formula: see text] having at most 2 vertices such that the induced subgraph [Formula: see text] is connected. We call a graph to be an i-2psd graph if it possesses an i-2psd set. In this article, we explore i-2psd graphs. We first provide complete structural characterization of separable i-2psd graphs and thereafter, in our quest to characterize i-2psd blocks, we characterize hexagon-free bipartite i-2psd blocks and exhibit a family of non-bipartite i-2psd blocks.