Linear forest
In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph, or a Disjoint [union of graphs|disjoint union] of nontrivial paths. Equivalently, it is an acyclic and claw-free graph. An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest. An undirected graph has graph invariant">graph (discrete mathematics)">graph invariant at most 1 if and only if it is a disjoint union of paths, i.e. it is linear. Any linear forest is a subgraph of the path graph with the same number of vertices.
Extensions to the notation
According to Habib and Peroche, a k-linear forest consists of paths of k or fewer nodes each.According to Burr and Roberts, an -linear forest has n vertices and j of its component paths have an odd number of vertices.
According to Faudree et al., a -linear or -linear forest has k edges, and t components of which s are single vertices; s is omitted if its value is not critical.
Derived concepts
The linear arboricity of a graph is the minimum number of linear forests into which the graph can be partitioned. For a graph of maximum degree, the linear arboricity is always at least, and it is conjectured that it is always at most.A linear coloring of a graph is a proper graph coloring in which the induced subgraph formed by each two colors is a linear forest. The linear chromatic number of a graph is the smallest number of colors used by any linear coloring. The linear chromatic number is at most proportional to, and there exist graphs for which it is at least proportional to this quantity.