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

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". This information, however, gets lost whenever $(T,t)$ contains vertices with label "prime". In this contribution, we aim at replacing "prime" vertices in $(T,t)$ by simple 0/1-labeled cycles, which leads to the concept of rooted labeled level-1 networks $(N,t)$. We characterize graphs that can be explained by such level-1 networks $(N,t)$, which generalizes the concept of graphs that can be explained by labeled trees, that is, cographs. We provide three novel graph classes: \emph{polar-cats} are a proper subclass of \emph{pseudo-cographs} which forms a proper subclass of \emph{prime polar-cats}. In particular, every cograph is a pseudo-cograph and prime polar-cats are precisely those graphs that can be explained by a labeled level-1 network. The class of prime polar-cats 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. In addition, we show under which conditions there is a unique least-resolved labeled level-1 network that explains a given graph and provide linear-time algorithms to recognize all these types of graphs and to construct level-1 networks to explain them.