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De novo genome assemblers assume the reference genome is unavailable, incomplete, highly fragmented, or significantly altered as in cancer tissues. Algorithms for de novo assembly have been developed to deal with and assemble a large number of short sequence reads from genome sequencing. In this review paper, we have provided an overview of the graph-theoretical side of de novo genome assembly algorithms. We have investigated the construction of fourteen graph data structures related to OLC-based and DBG-based algorithms in order to compare and discuss their application in different assemblers. In addition, the most significant and recent genome de novo assemblers are classified according to the extensive variety of original, generalized, and specialized versions of graph data structures.