In graph theory, a branch of mathematics, the linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two, i.e., a disjoint union of path graphs.
The linear arboricity of a graph
In a regular graph, the linear arboricity cannot equal
Linear arboricity is a variation of arboricity, the minimum number of forests that the edges of a graph can be partitioned into. Researchers have also studied linear k-arboricity, a variant of linear arboricity in which each path in the linear forest can have at most k edges. Unlike arboricity, which can be determined in polynomial time, linear arboricity is NP-hard. Even recognizing the graphs of linear arboricity two is NP-complete. However, for cubic graphs and other graphs of maximum degree three, the linear arboricity is always two, and a decomposition into two linear forests can be found in linear time using an algorithm based on depth-first search.