Hierarchical and Modularly-Minimal Vertex Colorings

Cographs are exactly the hereditarily well-colored graphs, i.e., the graphs for which a greedy vertex coloring of every induced subgraph uses only the minimally necessary number of colors $\chi(G)$. We show that greedy colorings are a special case of the more general hierarchical vertex colorings, which recently were introduced in phylogenetic combinatorics. Replacing cotrees by modular decomposition trees generalizes the concept of hierarchical colorings to arbitrary graphs. We show that every graph has a modularly-minimal coloring $\sigma$ satisfying $|\sigma(M)|=\chi(M)$ for every strong module $M$ of $G$. This, in particular, shows that modularly-minimal colorings provide a useful device to design efficient coloring algorithms for certain hereditary graph classes. For cographs, the hierarchical colorings coincide with the modularly-minimal coloring. As a by-product, we obtain a simple linear-time algorithm to compute a modularly-minimal coloring of $P_4$-sparse graphs.