A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices, while the weight of a set of vertices is the sum of their weights. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW(G). In what follows, all weights are real. Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition \(B_{X}\) and \(B_{Y}\). Then B is generating if there exists an independent set S such that \(S \cup B_{X}\) and \(S \cup B_{Y}\) are both maximal independent sets of G. Generating subgraphs play an important role in finding WCW(G). In the restricted case that a generating subgraph B is isomorphic to \(K_{1,1}\), its unique edge is called a relating edge. Deciding whether an input graph G is well-covered is co-NP-complete. Therefore finding WCW(G) is co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore, deciding whether a subgraph is generating is NP-complete as well. This article deals with graphs G such that \(\Delta (G)=|V(G)|-k\) for some \(k\in \mathbb {N}\). We prove that for this family recognizing well-covered graphs is a polynomial problem, while finding WCW(G) is co-NP-hard. To the best of our knowledge, this is the first family of graphs in the literature known to have these properties. For this set of graphs, recognizing relating edges and generating subgraphs is NP-complete. The article also deals with connected graphs for which \(\delta (G) = k\) or \(\delta (G) \ge \frac{k-1}{k}|V(G)|\). For these families of graphs recognizing well-covered graphs is co-NP-complete, while recognizing relating edges is NP-complete.

如果图G的所有最大独立集都具有相同的基数，则图 G是良好覆盖的。假设在其顶点上定义了权重函数w，而一组顶点的权重是它们的权重之和。如果所有最大独立集具有相同的权重，则G是w良好覆盖的。对于每个图G ，使得G被w良好覆盖的权重函数w的集合是一个向量空间，表示为WCW ( G )。以下所有重量均为实数。让B是G在二分\(B_{X}\)和\(B_{Y}\)的顶点集上的完整二部诱导子图。如果存在一个独立集合S使得\(S \cup B_{X}\)和\(S \cup B_{Y}\)都是G的最大独立集合，则B正在生成。生成子图在寻找WCW（G ）中起着重要作用。在生成子图B同构于\(K_{1,1}\) 的受限情况下，其唯一边称为相关边。判断输入图G是否被很好地覆盖了，并且是co-NP完全的。因此找到WCW ( G ) 是co-NP困难的。决定一条边是否相关是NP完全的。因此，判断子图是否正在生成也是NP完全的。本文处理图G，对于某些\(k\in \mathbb {N}\) ，满足 \ (\Delta (G)=|V(G)|-k\)。我们证明对于这个族来说，识别覆盖良好的图是一个多项式问题，而找到WCW ( G ) 是一个co-NP-难的。据我们所知，这是文献中已知的第一组具有这些属性的图。对于这组图，识别相关边并生成子图是NP完全的。本文还涉及\(\delta (G) = k\)或\(\delta (G) \ge \frac{k-1}{k}|V(G)|\) 的连通图。对于这些图族，识别良好覆盖的图是co-NP完全的，而识别相关边是NP完全的。