Linear forest

In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph,[1]: 200  or a disjoint union of nontrivial paths.[2]: 246  Equivalently, it is an acyclic and claw-free graph.[3]: 130, 131  An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest.[4]: 310 [5]: 107  An undirected graph has Colin de Verdière graph invariant at most 1 if and only if it is a (node-)disjoint union of paths, i.e. it is linear.[6]: 13, 19–21 [7]: 29, 35, 67 (3, 6, 29)  Any linear forest is a subgraph of the path graph with the same number of vertices.[8]: 55 

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