Let R=K[x1,…,xn] be the polynomial ring in n variables over a field
Nearby in the stack
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.