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Let $B$ be an induced complete bipartite subgraph of $G$ on vertex sets ofbipartition $B_{X}$ and $B_{Y}$. The subgraph $B$ is {\it generating} if thereexists an independent set $S$ such that each of $S \cup B_{X}$ and $S \cupB_{Y}$ 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. Thegraph $G$ is $w$-well-covered if and only if $w$ satisfies all restrictionsproduced by all generating subgraphs of $G$. Therefore, generating subgraphsplay an important role in characterizing weighted well-covered graphs. It is an\textbf{NP}-complete problem to decide whether a subgraph is generating, evenwhen the subgraph is isomorphic to $K_{1,1}$ \cite{bnz:related}. We present apolynomial algorithm for recognizing generating subgraphs for graphs withoutcycles of lengths 6 and 7.

Recognizing Generating Subgraphs in Graphs without Cycles of Lengths 6 and 7