linear forest
File:Linear forest.svg are not allowed as a subgraph (such as the claw in the second graph), and neither are cycles]]
In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph,{{Cite journal |last=Harary |first=Frank |author-link=Frank Harary |date=September 1970 |title=Covering and Packing in Graphs, I |journal=Annals of the New York Academy of Sciences |volume=175 |issue=1 |pages=198–205 |doi=10.1111/j.1749-6632.1970.tb56470.x |bibcode=1970NYASA.175..198H }}{{Rp|page=200}} or a disjoint union of nontrivial paths.{{Cite journal |last1=Burr |first1=Stefan A. |author-link=Stefan Burr |last2=Roberts |first2=John A. |date=May 1974 |title=On Ramsey numbers for linear forests |journal=Discrete Mathematics |publisher=North-Holland Publishing Company |volume=8 |issue=3 |pages=245–250 |doi=10.1016/0012-365x(74)90136-8 |mr=335325 }}{{Rp|page=246}} Equivalently, it is an acyclic and claw-free graph.{{Cite journal |last1=Brandstädt |first1=Andreas |author-link=Andreas Brandstädt |last2=Giakoumakis |first2=Vassilis |last3=Milanič |first3=Martin |date=2018-12-11 |title=Weighted efficient domination for some classes of H-free and of (H1,H2)-free graphs |journal=Discrete Applied Mathematics |publisher=Elsevier B.V. |volume=250 |pages=130–144 |doi=10.1016/j.dam.2018.05.012 |mr=3868662 |id={{EBSCOhost|45704539|132688071}} |doi-access=free }}{{Rp|pages=130,131}} An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest.{{Cite journal |last1=Enomoto |first1=Hikoe |last2=Péroche |first2=Bernard |date=Summer 1984 |title=The Linear Arboricity of Some Regular Graphs |journal=Journal of Graph Theory |volume=8 |issue=2 |pages=309–324 |doi=10.1002/jgt.3190080211 }}{{Rp|page=310}}{{Cite book |last1=Jain |first1=Sparsh |chapter-url=https://books.google.com/books?id=ZKJaEAAAQBAJ&dq=%22linear+forest%22+degree&pg=PA107 |title=Algorithms and Discrete Applied Mathematics: 8th International Conference, CALDAM 2022 |last2=Pallathumadam |first2=Sreejith K. |last3=Rajendraprasad |first3=Deepak |date=February 10–12, 2022 |publisher=Springer Nature |isbn=978-3-030-95017-0 |editor-last=Balachandran |editor-first=Niranjan |series=Lecture Notes in Computer Science |volume=13179 |location=Puducherry, India |publication-place=Cham, Switzerland |pages=103–114 |language=en |chapter=B0-VPG Representation of AT-free Outerplanar Graphs |doi=10.1007/978-3-030-95018-7_9 |editor-last2=Inkulu |editor-first2=R. }}{{Rp|page=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.{{Cite journal |last=de Verdière |first=Yves Colin |author-link=Yves Colin de Verdière |date=October 1990 |title=Sur un Nouvel Invariant des Graphes et un Critère de Planarité |journal=Journal of Combinatorial Theory, Series B |language=fr |publisher=Academic Press, Inc. |volume=50 |issue=1 |pages=11–21 |doi=10.1016/0095-8956(90)90093-F }}{{Rp|pages=13, 19-21}}{{Cite book |last1=van der Holst |first1=Hein |title=Graph Theory and Combinatorial Biology |last2=Lovász |first2=László |author-link2=László Lovász |last3=Schrijver |first3=Alexander |author-link3=Alexander Schrijver |date=1999 |publisher=János Bolyai Mathematical Society ({{langx|hu|Bolyai János Matematikai Társulat}}) |isbn=963-8022-90-6 |editor-last=Lovász |editor-first=László |series=Bolyai Society Mathematical Studies |volume=7 |publication-place=Budapest, Hungary |pages=29–85 [1–43] |chapter=The Colin de Verdière graph parameter |mr=1673503 |postscript=. Associated with the "International Colloquium on Combinatorics and Graph Theory" in Balatonlelle on July 1996 (see p. 5 and wwwold.sztaki.hu/library/publtop/gyarf.jhtml). |orig-date=Preliminary version, March 1997 |editor-last2=Gyárfás |editor-first2=András |editor-last3=Katona |editor-first3=Gyula |editor-last4=Recski |editor-first4=András |editor-last5=Székely |editor-first5=László |chapter-url=http://www.cs.elte.hu/~lovasz/colinsurv.ps |archive-url=https://web.archive.org/web/20070206001555/http://www.cs.elte.hu/~lovasz/colinsurv.ps |archive-date=2007-02-06 |url-status=dead}}{{Rp|pages=29, 35, 67 (3, 6, 29)}} Any linear forest is a subgraph of the path graph with the same number of vertices.{{Cite thesis |last=Clark |first=Curtis |title=An Approach to Graph Achievement Games: Ultimately Economical Graphs |date=1984 |degree=PhD |publisher=University of Michigan |id={{ProQuest|303324911}} (UMI 8502782) |place=Ann Arbor, Michigan |isbn=979-8-204-34535-5 |via=ProQuest Dissertations & Theses Global}}{{Rp|page=55}}
Extensions to the notation
According to Habib and Peroche, a k-linear forest consists of paths of k or fewer nodes each.{{Cite journal |last1=Habib |first1=M. |last2=Peroche |first2=B. |date=1982 |title=Some problems about linear arboricity |journal=Discrete Mathematics |publisher=North-Holland Publishing Company |volume=41 |issue=2 |pages=219–220 |doi=10.1016/0012-365x(82)90209-6 |doi-access=free }}{{Rp|page=219}}
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.{{Rp|pages=245, 246}}
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.{{Cite journal |last1=Faudree |first1=Ralph J. |author-link=Ralph Faudree |last2=Gould |first2=Ronald J. |author-link2=Ronald Gould (mathematician) |last3=Jacobson |first3=Michael S. |author-link3=Michael Scott Jacobson |date=28 March 2009 |title=Pancyclic graphs and linear forests |journal=Discrete Mathematics |publisher=Elsevier B.V. |volume=309 |issue=5 |pages=1178–1189 |doi=10.1016/j.disc.2007.12.094 |doi-access=free }}{{Rp|pages=1178, 1179}}
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 .{{citation
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| doi = 10.1007/BF02783300 | doi-access = free
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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.{{citation
| last = Yuster | first = Raphael | author-link = Raphael Yuster
| doi = 10.1016/S0012-365X(97)00209-4
| issue = 1–3
| journal = Discrete Mathematics
| mr = 1614290
| pages = 293–297
| title = Linear coloring of graphs
| volume = 185
| year = 1998| doi-access = free
}}.