On invariants of hereditary graph properties Original Research Article

The product image of graph properties image is the class of all graphs having a vertex-partition into two parts inducing subgraphs with properties image and image, respectively. For a graph invariant image and a graph property image we define image as the minimum of image taken over all minimal forbidden subgraphs F of image. An invariant of graph properties image is said to be additive with respect to reducible hereditary properties if image for every pair of hereditary properties image. In this paper, we provide necessary and sufficient conditions for invariants to be additive with respect to reducible hereditary graph properties. We prove that the subchromatic number, the degeneracy number and tree-width and some other invariants of hereditary graph properties satisfy those conditions.