From Modular Decomposition Trees to Level-1 Networks: Pseudo-Cographs, Polar-Cats and Prime Polar-Cats

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} ∈ E(G) if and only if the lowest common ancestor lcaN(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 pseudocographs which forms a proper subclass of prime. In particular, every cograph is a pseudo-cograph and prime is precisely the class of graphs the can be explained by a labeled level-1 network. The class prime 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.