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 \({{\mathcal {S}}}\) 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 \({\mathcal {S}}\). These results seem to be harmonic with respect to other polynomial results for WIS on [subclasses of] certain \(S_{i,j,k}\)-free graphs and to other structure results on [subclasses of] Triangle-free graphs.

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的不相交并集。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 a \ (P_4 \)是四个顶点的诱导路径，三角形是三个顶点的循环，特别是结果证明 (ii) 对于每个 ( \ (P_4 + P_4 \) , Triangle) - 自由图G有一个族\ ({{\ mathcal {S}}} \) V ( G )的子集归纳出G 的（完全）二部子图，其中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 的每个最大独立集是包含在\ ({\ mathcal {S}} \) 的某个成员中。这些结果似乎与 WIS 在 [子类] 某些\ (S_ {i, j, k} \)上的其他多项式结果一致无三角形图的 [子类] 上的无图和其他结构结果。