A graph $G$ is well-covered if all maximal independent sets are of the same cardinality. Let $w:V(G) \longrightarrow\mathbb{R}$ be a weight function. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. An edge $xy \in E(G)$ is relating if there exists an independent set $S$ such that both $S \cup \{x\}$ and $S \cup \{y\}$ are maximal independent sets in the graph. If $xy$ is relating then $w(x)=w(y)$ for every weight function $w$ such that $G$ is $w$-well-covered. Relating edges play an important role in investigating $w$-well-covered graphs. The decision problem whether an edge in a graph is relating is NP-complete. We prove that the problem remains NP-complete when the input is restricted to graphs without cycles of length $6$. This is an unexpected result because recognizing relating edges is known to be polynomially solvable for graphs without cycles of lengths $4$ and $6$, graphs without cycles of lengths $5$ and $6$, and graphs without cycles of lengths $6$ and $7$. A graph $G$ belongs to the class $W_2$ if every two pairwise disjoint independent sets in $G$ are included in two pairwise disjoint maximum independent sets. It is known that if $G$ belongs to the class $W_2$, then it is well-covered. A vertex $v \in V(G)$ is shedding if for every independent set $S \subseteq V(G)-N[v]$, there exists a vertex $u \in N(v)$ such that $S \cup \{u\}$ is independent. Shedding vertices play an important role in studying the class $W_2$. Recognizing shedding vertices is co-NP-complete, even when the input is restricted to triangle-free graphs. We prove that the problem is co-NP-complete for graphs without cycles of length $6$.

翻译：图$G$若所有极大独立集具有相同基数，则称为良好覆盖图。设$w:V(G) \longrightarrow\mathbb{R}$为权函数，若所有极大独立集具有相同权重，则$G$为$w$-良好覆盖图。边$xy \in E(G)$被称为关联边，若存在独立集$S$使得$S \cup \{x\}$和$S \cup \{y\}$均为图的极大独立集。若$xy$是关联边，则对于使$G$为$w$-良好覆盖图的任意权函数$w$，有$w(x)=w(y)$。关联边在研究$w$-良好覆盖图时具有重要作用。判定图中某边是否为关联边的决策问题是NP完全的。我们证明即使输入限制于不含6圈长度的图，该问题仍保持NP完全性。这一结果出乎意料，因为已知对于不含4圈和6圈、不含5圈和6圈、以及不含6圈和7圈的图类，关联边识别可在多项式时间内求解。若图$G$中任意两两不相交的独立集均可包含于两个两两不相交的极大独立集中，则称$G$属于$W_2$类。已知若$G$属于$W_2$类，则其为良好覆盖图。顶点$v \in V(G)$被称为脱落顶点，若对于任意独立集$S \subseteq V(G)-N[v]$，存在顶点$u \in N(v)$使得$S \cup \{u\}$仍为独立集。脱落顶点在研究$W_2$类中起关键作用。识别脱落顶点是co-NP完全的，即使输入限制于无三角形图时也是如此。我们证明该问题对于不含6圈长度的图仍保持co-NP完全性。