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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 G\N[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.