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| {{about|graph theory|the shrub|Plumeria alba}}
| | НАША КОМАНДА<br><br>КОНСТАНТИН<br>дизайнер <br><br>Я был на Вашем месте, я тоже заказывал [http://kks.by дизайн студия минск]. Обращайтесь и я Вас удивлю!<br><br>ЕКАТЕРИНА <br>режиссер <br><br>Я арт-менеджер и мое любимое дело - это фото-режиссура.<br><br><br>РИЧАРД<br>кот <br><br>Я символ студии и я слежу за работой Кати и Кости.<br><br>http://www.kks.by<br><br>Feel free to visit my web site - [http://kks.by/?cat=14 модная музыка] |
| [[File:Caterpillar tree.svg|thumb|300px|A caterpillar]]
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| In [[graph theory]], a '''caterpillar''' or '''caterpillar tree''' is a [[tree (graph theory)|tree]] in which all the vertices are within distance 1 of a central path.
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| Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by A. Hobbs.<ref name="hs73"/><ref name="eb87"/> As {{harvtxt|Harary|Schwenk|1973}} colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed."<ref name="hs73">{{citation
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| | last1 = Harary | first1 = Frank | author1-link = Frank Harary
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| | last2 = Schwenk | first2 = Allen J.
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| | issue = 4
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| | journal = Discrete Mathematics
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| | pages = 359–365
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| | title = The number of caterpillars
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| | volume = 6
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| | year = 1973}}.</ref>
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| ==Equivalent characterizations==
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| The following characterizations all describe the caterpillar trees:
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| *They are the trees for which removing the leaves and incident edges produces a [[path graph]].<ref name="eb87"/><ref name="hs71"/>
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| *They are the trees in which there exists a path that contains every node of degree two or more.
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| *They are the trees in which every node of degree at least three has at most two non-leaf neighbors.
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| *They are the trees that do not contain as a subgraph the graph formed by replacing every edge in the [[star graph]] ''K''<sub>1,3</sub> by a path of length two.<ref name="hs71"/>
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| *They are the connected graphs that can be [[graph drawing|drawn]] with their vertices on two parallel lines, with edges represented as non-crossing line segments that have one endpoint on each line.<ref name="hs71"/><ref>{{citation
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| | last1 = Harary | first1 = Frank | author1-link = Frank Harary
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| | last2 = Schwenk | first2 = Allen J.
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| | journal = Utilitas Math.
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| | pages = 203–209
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| | title = A new crossing number for bipartite graphs
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| | volume = 1
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| | year = 1972}}.</ref>
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| *They are the trees whose [[Glossary of graph theory#Distance|square]] is a [[Hamiltonian graph]]. That is, in a caterpillar, there exists a cyclic sequence of all the vertices in which each adjacent pair of vertices in the sequence is at distance one or two from each other, and trees that are not caterpillars do not have such a sequence. A cycle of this type may be obtained by drawing the caterpillar on two parallel lines and concatenating the sequence of vertices on one line with the reverse of the sequence on the other line.<ref name="hs71">{{citation
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| | last1 = Harary | first1 = Frank | author1-link = Frank Harary
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| | last2 = Schwenk | first2 = Allen J.
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| | doi = 10.1112/S0025579300008494
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| | journal = Mathematika
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| | pages = 138–140
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| | title = Trees with Hamiltonian square
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| | volume = 18
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| | year = 1971}}.</ref>
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| *They are the trees whose [[line graph]]s contain a [[Hamiltonian path]]; such a path may be obtained by the ordering of the edges in a two-line drawing of the tree. More generally the number of edges that need to be added to the line graph of an arbitrary tree so that it contains a Hamiltonian path (the size of its [[Hamiltonian completion]]) equals the minimum number of edge-disjoint caterpillars that the edges of the tree can be decomposed into.<ref>{{citation
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| | last = Raychaudhuri | first = Arundhati
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| | doi = 10.1016/0020-0190(95)00163-8
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| | issue = 6
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| | journal = [[Information Processing Letters]]
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| | pages = 299–306
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| | title = The total interval number of a tree and the Hamiltonian completion number of its line graph
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| | volume = 56
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| | year = 1995}}.</ref>
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| *They are the connected graphs of [[pathwidth]] one.<ref name="pt99"/>
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| *They are the connected [[triangle-free graph|triangle-free]] [[interval graph]]s.<ref>{{citation
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| | last = Eckhoff | first = Jürgen
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| | doi = 10.1002/jgt.3190170112
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| | issue = 1
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| | journal = Journal of Graph Theory
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| | pages = 117–127
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| | title = Extremal interval graphs
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| | volume = 17
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| | year = 1993}}.</ref>
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| ==Generalizations==
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| A ''k''-tree is a [[chordal graph]] with exactly {{nowrap|''n'' − ''k''}} [[maximal clique]]s, each containing {{nowrap|''k'' + 1}} vertices; in a ''k''-tree that is not itself a {{nowrap|(''k'' + 1)-clique}}, each maximal clique either separates the graph into two or more components, or it contains a single leaf vertex, a vertex that belongs to only a single maximal clique. A ''k''-path is a ''k''-tree with at most two leaves, and a ''k''-caterpillar is a ''k''-tree in which the non-leaf vertices [[induced subgraph|induce]] a ''k''-path. In this terminology, a 1-caterpillar is the same thing as a caterpillar tree, and ''k''-caterpillars are the edge-maximal graphs with [[pathwidth]] ''k''.<ref name="pt99">{{citation
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| | last1 = Proskurowski | first1 = Andrzej
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| | last2 = Telle | first2 = Jan Arne
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| | journal = Discrete Mathematics and Theoretical Computer Science
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| | pages = 167–176
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| | title = Classes of graphs with restricted interval models
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| | url = http://www.emis.ams.org/journals/DMTCS/volumes/abstracts/pdfpapers/dm030404.pdf
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| | volume = 3
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| | year = 1999}}.</ref>
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| A '''lobster''' graph is a [[tree (graph theory)|tree]] in which all the vertices are within distance 2 of a central [[path (graph theory)|path]].<ref>{{mathworld|urlname=Lobster|title=Lobster}}</ref>
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| ==Enumeration==
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| Caterpillars provide one of the rare [[graph enumeration]] problems for which a precise formula can be given: when ''n'' ≥ 3, the number of caterpillars with ''n'' unlabeled vertices is <ref name="hs73"/>
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| :<math>2^{n-4}+2^{\lfloor (n-4)/2\rfloor}.</math>
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| For ''n'' = 1, 2, 3, ... the numbers of ''n''-vertex caterpillars are
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| :1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, ... {{OEIS|A005418}}.
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| ==Computational Complexity==
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| Finding a spanning caterpillar in a graph
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| is [[NP-complete]]. A related optimization problem is the Minimum Spanning
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| Caterpillar Problem (MSCP) , where a graph has dual costs
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| over its edges and the goal is to find a caterpillar tree that spans the
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| input graph and has the smallest overall cost. Here the cost of the caterpillar is defined as the sum
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| of the costs of its edges, where each edge takes one of the two costs based on its role
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| as a leaf edge or an internal one. There is no f(n)-[[approximation algorithm]] for the MSCP unless [[P = NP]].
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| Here f(n) is any polynomial-time computable function of n, the number of nodes of
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| a graph.<ref name="mk11"/>
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| There is a parametrized algorithm that finds an optimal solution for the MSCP in
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| bounded [[treewidth]] graphs. So both the Spanning Caterpillar Problem and the MSCP have
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| linear time algorithms if a graph is an outerplanar, a series-parallel, or a [[Halin graph]].
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| <ref name="mk11">{{cite thesis |type=Ph.D. |first=Masoud |last=Khosravani |title=Searching for optimal caterpillars in general and bounded treewidth graphs |publisher=University of Auckland |year=2011| url = https://researchspace.auckland.ac.nz/handle/2292/8360?show=full}}</ref>
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| ==Applications==
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| Caterpillar trees have been used in [[chemical graph theory]] to represent the structure of [[benzenoid]] [[hydrocarbon]] molecules. In this representation, one forms a caterpillar in which each edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the corresponding rings belong to a sequence of rings connected end-to-end in the structure. {{harvtxt|El-Basil|1987}} writes, "It is amazing that nearly all graphs that played an important role in what is now called "chemical graph theory" may be related to caterpillar trees." In this context, caterpillar trees are also known as '''benzenoid trees''' and '''Gutman trees''', after the work of Ivan Gutman in this area.<ref name="eb87">{{citation
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| | last = El-Basil | first = Sherif
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| | doi = 10.1007/BF01205666
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| | issue = 2
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| | journal = Journal of Mathematical Chemistry
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| | pages = 153–174
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| | title = Applications of caterpillar trees in chemistry and physics
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| | volume = 1
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| | year = 1987}}.</ref><ref>{{citation
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| | last = Gutman | first = Ivan
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| | doi = 10.1007/BF00554539
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| | issue = 4
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| | journal = Theoretica Chimica Acta
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| | pages = 309–315
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| | title = Topological properties of benzenoid systems
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| | volume = 45
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| | year = 1977}}.</ref><ref>{{citation
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| | last = El-Basil | first = Sherif
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| | contribution = Caterpillar (Gutman) trees in chemical graph theory
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| | doi = 10.1007/3-540-51505-4_28
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| | editor1-last = Gutman | editor1-first = I.
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| | editor2-last = Cyvin | editor2-first = S. J.
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| | pages = 273–289
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| | series = Topics in Current Chemistry
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| | title = Advances in the Theory of Benzenoid Hydrocarbons
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| | volume = 153
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| | year = 1990}}.</ref>
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{mathworld|urlname=Caterpillar|title=Caterpillar}}
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| [[Category:Trees (graph theory)]]
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| [[Category:Mathematical chemistry]]
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