We characterize graphs $G$ that can be explained by rooted labeled level-1 networks $(N,t)$, i.e., $N$ is equipped with a binary vertex-labeling $t$ such that $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_N(x,y)$ of $x$ and $y$ has label ``$1$''. This generalizes the concept of graphs that can be explained by labeled trees, that is, cographs. We provide three novel graph classes: polar-cats are a proper subclass of pseudo-cographs which forms a proper subclass of \PrimeCat. In particular, every cograph is a pseudo-cograph and \PrimeCat is precisely the class of graphs the can be explained by a labeled level-1 network. The class \PrimeCat is defined in terms of the modular decomposition of graphs and the property that all prime modules ``induce'' polar-cats. We provide a plethora of structural results and characterizations for graphs of these new classes and give linear-time algorithms to recognize them and to construct level-1 networks to explain them.
翻译：我们用有根标签的等级-1网络来解释$G$, 即$(n,t), 美元, 也就是说, $, 美元, 配有双面脊椎标签, 美元, 也就是说, $x, y ⁇ in E(G), 美元, 特别是, 每一个最普通的祖先 $\ mathrm{lca ⁇ N(x,y) $ 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元。 这概括了可以用有标签的树木来解释的图表概念。 我们提供了三种新型图表的模块解析和属性: 极地猫是一个适当的伪造图小类子类, 构成适当的\ Pricat 子类。 特别是, 每部的cophical is a protographical and licalalals, 我们向这些图表的模型和直线级解算结果提供。
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