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The notion of f-graphs and f-ideals are relatively new and have been studied in many papers. In this paper, we have generalized the idea of f-graphs and f-ideals to quasi f-graphs and quasi f-ideals, respectively. We have characterized all quasi f-graphs and quasi f-ideals of degree 2 and determined all the minimal primes ideals of these ideals. Furthermore, construction of quasi f-ideals of degree 2 has been described; the formula for computing Hilbert function and Hilbert series of the polynomial ring modulo the edge ideal of the quasi f-graph has been provided.