We revisit a classical paper about (even hole, triangle)-free graphs [Conforti, Cornuéjols, Kapoor and Vu\v sković, Triangle-free graphs that are signable without even holes, Journal of Graph Theory, 34(3), 204--220, 2000]. In fact, the previous study describes a more general class, the so called triangle-free odd signable graphs, and we further generalise the class to the (theta, triangle, wac)-free graphs (not worth defining in an abstract). We exhibit a stronger structure theorem, by precisely describing basic classes and separators. We prove that the separators preserve the treewidth and several properties. Some consequences are a recognition algorithm with running time $O(|V(G)|^4|E(G)|)$, a proof that the treewidth of graphs in the class is at most~4 (improving a previous bound of~5), and a simple criterion to decide if a graph in the class is planar.

翻译：我们重新审视一篇关于(偶洞，三角形)自由图的经典论文[Conforti, Cornuéjols, Kapoor and Vušković, Triangle-free graphs that are signable without even holes, Journal of Graph Theory, 34(3), 204--220, 2000]。实际上，先前的研究描述了一个更一般的类，即所谓的无三角形奇可符号图，我们进一步将该类推广到(theta，三角形，wac)自由图（无需在摘要中定义）。我们通过精确描述基本类和分离子，展示了一个更强的结构定理。我们证明了分离子保持了树宽和若干性质。一些推论包括运行时间为$O(|V(G)|^4|E(G)|)$的识别算法，该类中图的树宽至多为4（改进了先前的5的上界），以及判断该类中图是否为平面图的一个简单准则。