  Let $B$ be an induced complete bipartite subgraph of $G$ on vertex sets of
bipartition $B_{X}$ and $B_{Y}$. The subgraph $B$ is {\it generating} if there
exists an independent set $S$ such that each of $S \cup B_{X}$ and $S \cup
B_{Y}$ is a maximal independent set in the graph. If $B$ is generating, it
\textit{produces} the restriction $w(B_{X})=w(B_{Y})$. Let $w:V(G)
\longrightarrow\mathbb{R}$ be a weight function. We say that $G$ is
$w$-well-covered if all maximal independent sets are of the same weight. The
graph $G$ is $w$-well-covered if and only if $w$ satisfies all restrictions
produced by all generating subgraphs of $G$. Therefore, generating subgraphs
play an important role in characterizing weighted well-covered graphs. It is an
\textbf{NP}-complete problem to decide whether a subgraph is generating, even
when the subgraph is isomorphic to $K_{1,1}$ \cite{bnz:related}. We present a
polynomial algorithm for recognizing generating subgraphs for graphs without
cycles of lengths 6 and 7.
