The notion of f-ideal was introduced in [G. Q. Abbasi, S. Ahmad, I. Anwar and W. A. Baig, f-Ideals of degree 2, Algebra Collog. 19(1) (2012) 921-926] in 2012 and has been studied in many papers after that. In this paper, we have studied those graphs whose Stanley-Reisner ideals turn out to be f-ideals. We give a characterization and construction of these graphs and show that, unlike f-graphs, these graphs are always connected. We have also discussed when these graphs are complete bipartite graphs. Moreover, we classify those graphs for which both the facet ideal (the edge ideal) and the Stanley-Resiner ideal, are f-ideals.
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