Vertex arboricity of cographs
Sebasti\'an Gonz\'alez Hermosillo de la Maza, Pavol Hell, C\'esar, Hern\'andez Cruz, Seyyed Aliasghar Hosseini, Payam Valadkhan

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 and…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Limits and Structures in Graph Theory
