  A graph is well-covered if all its maximal independent sets are of the same
cardinality (Plummer, 1970). If G is a well-covered graph, has at least two
vertices, and G-v is well-covered for every vertex v, then G is a
1-well-covered graph (Staples, 1975). We call G a {\lambda}-quasi-regularizable
graph if {\lambda} |S| =< |N(S)| for every independent set S of G. The
independence polynomial I(G;x) is the generating function of independent sets
in a graph G (Gutman & Harary, 1983). The Roller-Coaster Conjecture (Michael &
Travis, 2003), saying that for every permutation {\sigma} of the set
{({\alpha}/2),...,{\alpha}} there exists a well-covered graph G with
independence number {\alpha} such that the coefficients (s_{k}) of I(G;x) are
chosen in accordance with {\sigma}, has been validated in (Cutler & Pebody,
2017). In this paper, we show that independence polynomials of
{\lambda}-quasi-regularizable graphs are partially unimodal. More precisely,
the coefficients of an upper part of I(G;x) are in non-increasing order. Based
on this finding, we prove that the domain of the Roller- Coaster Conjecture can
be shortened for well-covered graphs and 1-well-covered graphs.
