Partitioning Cographs into Two Forests and One Independent Set

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 $p=2$.

Our result gives a concrete forbidden induced subgraph characterization of cographs that can be partitioned into two forests and one independent set. Since our proof is algorithmic, we can apply our characterization to complement the recognition algorithm for partitionable cographs by an algorithm to certify non-partitionable cographs by finding a forbidden induced subgraph.

This research was supported by the first author’s NSERC Discovery Grant and the second author’s SEP-CONACYT grant A1-S-8397.