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] or a disjoint union of nontrivial paths.[2] Equivalently, it is an acyclic and claw-free graph.[3] An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest.[4][5] 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][7] Any linear forest is a subgraph of the path graph with the same number of vertices.[8]
Extensions to the notation
According to Habib and Peroche, a k-linear forest consists of paths of k or fewer nodes each.[9]
According to Burr and Roberts, an (n, j)-linear forest has n vertices and j of its component paths have an odd number of vertices.[2]
According to Faudree et al., a (k, t)-linear or (k, t, s)-linear forest has k edges, and t components of which s are single vertices; s is omitted if its value is not critical.[10]
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 .[11]
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.[12]
References
- ↑ Harary, Frank (September 1970), "Covering and Packing in Graphs, I", Annals of the New York Academy of Sciences 175 (1): 198–205, doi:10.1111/j.1749-6632.1970.tb56470.x, Bibcode: 1970NYASA.175..198H
- ↑ 2.0 2.1 Burr, Stefan A.; Roberts, John A. (May 1974), "On Ramsey numbers for linear forests", Discrete Mathematics (North-Holland Publishing Company) 8 (3): 245–250, doi:10.1016/0012-365x(74)90136-8
- ↑ Brandstädt, Andreas; Giakoumakis, Vassilis; Milanič, Martin (2018), "Weighted efficient domination for some classes of H-free and of (H1,H2)-free graphs", Discrete Applied Mathematics (Elsevier B.V.) 250: 130–144, doi:10.1016/j.dam.2018.05.012, EBSCOhost 45704539, 132688071
- ↑ Enomoto, Hikoe; Péroche, Bernard (1984), "The Linear Arboricity of Some Regular Graphs", Journal of Graph Theory 8 (2): 309–324, doi:10.1002/jgt.3190080211
- ↑ Jain, Sparsh; Pallathumadam, Sreejith K.; Rajendraprasad, Deepak (February 10–12, 2022), "B0-VPG Representation of AT-free Outerplanar Graphs", written at Puducherry, India, in Balachandran, Niranjan (in en), Algorithms and Discrete Applied Mathematics: 8th International Conference, CALDAM 2022, Lecture Notes in Computer Science, 13179, Cham, Switzerland: Springer Nature, pp. 103–114, doi:10.1007/978-3-030-95018-7_9, ISBN 978-3-030-95017-0, https://books.google.com/books?id=ZKJaEAAAQBAJ&dq=%22linear+forest%22+degree&pg=PA107
- ↑ de Verdière, Yves Colin (October 1990), "Sur un Nouvel Invariant des Graphes et un Critère de Planarité" (in fr), Journal of Combinatorial Theory, Series B (Academic Press, Inc.) 50 (1): 11–21, doi:10.1016/0095-8956(90)90093-F
- ↑ van der Holst, Hein; Lovász, László; Schrijver, Alexander (1999), "The Colin de Verdière graph parameter", in Lovász, László, Graph Theory and Combinatorial Biology, Bolyai Society Mathematical Studies, 7, Budapest, Hungary: János Bolyai Mathematical Society, pp. 29–85, ISBN 963-8022-90-6. Preliminary version, March 1997; see pp. 29, 35, 67 (pp. 3, 6, 29 of preliminary version)
- ↑ Clark, Curtis (1984), An Approach to Graph Achievement Games: Ultimately Economical Graphs, Ann Arbor, Michigan: University of Michigan, p. 55, ProQuest 303324911 (UMI 8502782), ISBN 979-8-204-34535-5
- ↑ Habib, M.; Peroche, B. (1982), "Some problems about linear arboricity", Discrete Mathematics (North-Holland Publishing Company) 41 (2): 219–220, doi:10.1016/0012-365x(82)90209-6
- ↑ Faudree, Ralph J.; Gould, Ronald J.; Jacobson, Michael S. (28 March 2009), "Pancyclic graphs and linear forests", Discrete Mathematics (Elsevier B.V.) 309 (5): 1178–1189, doi:10.1016/j.disc.2007.12.094
- ↑ "The linear arboricity of graphs", Israel Journal of Mathematics 62 (3): 311–325, 1988, doi:10.1007/BF02783300.
- ↑ "Linear coloring of graphs", Discrete Mathematics 185 (1–3): 293–297, 1998, doi:10.1016/S0012-365X(97)00209-4.
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