  A graph is well-covered if every maximal independent set has the same
cardinality. The recognition problem of well-covered graphs is known to be
co-NP-complete. Let w be a weight function defined on the vertices of G. Then G
is w-well-covered if all maximal independent sets of G are of the same weight.
The set of weight functions w for which a graph is w-well-covered is a vector
space. We prove that finding the vector space of weight functions under which
an input graph is w-well-covered can be done in polynomial time, if the input
graph does not contain cycles of length 4, 5, 6 and 7.
