Hierarchical and modularly-minimal vertex colorings

Cographs are exactly the hereditarily well-colored graphs, i.e., graphs for which a greedy vertex coloring of every induced subgraph uses only minimally necessary number colors χ(G). Greedy colorings shown to be subclass hierarchical that naturally appear in phylogenetic combinatorics. The cographs form special class minimal colorings. fact cotrees modular decomposition trees suggests natural generalization: A σ is modularly-minimal if |σ(M)| = χ(M) strong module M G. We show graph admits modularly coloring. On cographs, furthermore, coincide with For certain hereditary classes efficient algorithms can designed compute