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Let r = k[x1, . . . , xn] be the polynomial ring in n variables over a field k. we show that if g is a connected graph with a basic 5-cycle c, then g is a sequentially cohen-macaulay graph if and only if there exists a shedding vertex x of c such that g x and gn[x] are sequentially cohen- macaulay graphs. furthermore, we study the sequentially cohen-macaulay and castelnuovo-mumford regularity of square-free monomial ideals in some special cases.