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Description: Sparse graphs are ubiquitous in today's world, and have been of interest to a wide group of researchers across disciplines. With the rapid growth in the size of data in recent times, it has become imperative that we make use of the sparsity of datasets to improve performance. Moreover, sparse graphs can often not be studied with the same tools one uses to decipher dense graphs. The study of sparse graphs is thus a rich and exciting field. We present some completed work, and a series of open problems we hope to answer regarding sparse graphs; some more theoretical, some with explicit use cases. The technique at the heart of our (diverse) approaches are decompositions. We study different ways in which sparse graphs may be decomposed, and how different decompositions can be used to answer various pertinent questions.