Abstract
We consider a variation of arboricity, where a graph is partitioned into p forests and q independent sets. These problems are NP-complete in general, but polynomial-time solvable in the class of cographs; in fact, for each p and q there are only finitely many minimal non-partitionable cographs. In previous investigations it was revealed that when \(p=0\) or \(p=1\), these minimal non-partitionable cographs can be uniformly described as one family of obstructions valid for all values of q. We investigate the next case, when \(p=2\); we provide the complete family of minimal obstructions for \(p=2, q=1\), and find that they include more than just the natural extensions of the previously described obstructions for \(p=2, q=0\). Thus a uniform description for all q seems unlikely already in the case \(p=2\).
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