(a, b)-decomposition

In graph theory, the (a, b)-decomposition of an undirected graph is a partition of its edges into a + 1 sets, each one of them inducing a forest, except one which induces a graph with maximum degree b. If this graph is also a forest, then we call this a  F(a, b)-decomposition.

A graph with arboricity a is (a, 0)-decomposable. Every (a, 0)-decomposition or (a, 1)-decomposition is a F(a, 0)-decomposition or a F(a, 1)-decomposition respectively.

Graph classes

 * Every planar graph is F(2, 4)-decomposable.
 * Every planar graph $$G$$ with girth at least $$g$$ is
 * F(2, 0)-decomposable if $$g \ge 4$$.
 * (1, 4)-decomposable if $$g \ge 5$$.
 * F(1, 2)-decomposable if $$g \ge 6$$.
 * F(1, 1)-decomposable if $$g \ge 8$$, or if every cycle of $$G$$ is either a triangle or a cycle with at least 8 edges not belonging to a triangle.
 * (1, 5)-decomposable if $$G$$ has no 4-cycles.
 * Every outerplanar graph is F(2, 0)-decomposable and (1, 3)-decomposable.