Vertex arboricity of cographs

TL;DR

This paper studies the arboricity of cographs, providing a complete characterization for arboricity two, bounds for higher arboricities, and a polynomial-time algorithm for a generalized vertex partition problem.

Contribution

It offers a complete list of minimal obstructions for arboricity two in cographs, bounds for higher cases, and a versatile polynomial-time algorithm for related partition problems.

Findings

Complete list of minimal cograph obstructions for arboricity two

Bounds on size and height of minimal obstructions for higher arboricities

Polynomial-time algorithm for generalized vertex partition problem

Abstract

Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for the arboricity we want to partition the vertices into cycle-free subsets (i.e., forests). Arboricity is NP-hard in general, and our focus is on the arboricity of cographs. For arboricity two, we obtain the complete list of minimal cograph obstructions. These minimal obstructions do generalize to higher arboricities; however, we no longer have a complete list, and in fact, the number of minimal cograph obstructions grows exponentially with arboricity. We obtain bounds on their size and the height of their cotrees. More generally, we consider the following common generalization of colouring and partition into forests: given non-negative integers $p$ and $q$, we ask if a given cograph $G$ admits a vertex partition into $p$ forests and $q$ independent sets. We give a polynomial-time dynamic programming algorithm for this problem. In fact, the algorithm solves a more general problem which also includes several other problems such as finding a maximum $q$-colourable subgraph, maximum subgraph of arboricity-$p$, minimum vertex feedback set and minimum $q$ of a $q$-colourable vertex feedback set.