The notion of [Formula: see text]-ideal was introduced in [G. Q. Abbasi, S. Ahmad, I. Anwar and W. A. Baig, [Formula: see text]-Ideals of degree 2, Algebra Colloq. 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 [Formula: see text]-ideals. We give a characterization and construction of these graphs and show that, unlike [Formula: see text]-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 [Formula: see text]-ideals.
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.
BackgroundBecause of the anatomical properties of the inferior calyx, lower pole stones are difficult to remove through the ureter, even if the stones are fragmented. Retrograde intra-renal surgery (RIRS) is typically employed to treat the smaller lower pole stones (1.0-2.0 cm) while percutaneous nephrolithotomy (PCNL) is primarily used to treat the larger diameter stones or when RIRS has failed to clear the stones. This study was conducted to compare mini-PCNL and RIRS for the management of lower pole kidney stones in terms of stone clearance.
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