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On the vertex-arboricity of planar graphs. (English) Zbl 1144.05024
The vertex-arboricity $$\alpha(G)$$ of a graph $$G$$ is the minimum number of parts in a partition of the vertices so that each part induces a forest. It is known that $$\alpha(G) \leq 3$$ for any planar graph $$G$$. In this paper the authors show that $$\alpha(G) \leq 2$$ whenever $$G$$ is planar and has no 4-cycles. They also show that $$\alpha(G) \leq 2$$ if $$G$$ is planar and any two triangles are of distance at least 3. The paper contains several nice conjectures about vertex-arboricity.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
vertex-arboricity
Full Text:
##### References:
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