The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs $G$ and $H$, $G+H$ denotes the disjoint union of $G$ and $H$. This manuscript shows that (i) WIS can be solved for ($P_4+P_4$, Triangle)-free graphs in polynomial time, where a $P_4$ is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every ($P_4+P_4$, Triangle)-free graph $G$ there is a family ${\cal S}$ of subsets of $V(G)$ inducing (complete) bipartite subgraphs of $G$, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of $G$ is contained in some member of ${\cal S}$. These results seem to be harmonic with respect to other polynomial results for WIS on certain [subclasses of] $S_{i,j,k}$-free graphs and to other structure results on [subclasses of] Triangle-free graphs.

中文翻译：

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($P_4+P_4$,Triangle)-free 图中的独立集合

最大权重独立集问题 (WIS) 是众所周知的 NP 难题。研究 WIS 的一种流行方法是检测可以在多项式时间内求解 WIS 的图类，特别是参考遗传图类，即由遗传图属性定义或等效地通过禁止一个或多个诱导子图来定义。给定两个图 $G$ 和 $H$，$G+H$ 表示 $G$ 和 $H$ 的不相交联合。这份手稿表明 (i) WIS 可以在多项式时间内解决 ($P_4+P_4$, Triangle)-free 图，其中 $P_4$ 是四个顶点的诱导路径，三角形是三个顶点的循环，并且特别是结果证明 (ii) 对于每个 ($P_4+P_4$, Triangle)-free 图 $G$ 有一个 $V(G)$ 的子集族 ${\cal S}$完整）$G$ 的二部子图，它包含多项式多个成员并且可以在多项式时间内计算，使得 $G$ 的每个最大独立集都包含在 ${\cal S}$ 的某个成员中。这些结果似乎与 WIS 在某些 [子类] $S_{i,j,k}$-free 图上的其他多项式结果和其他结构结果在 [子类] 无三角形图上是和谐的。