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| In [[graph theory]], a '''connected dominated set''' and a '''maximum leaf spanning tree''' are two closely related structures defined on an [[undirected graph]].
| | 38 year-old Electronic Device Deals Staff (Standard ) Cameron Kull from Timmins, likes r/c cars, [http://www.healthymonadnock.org/activity/p/5083/ industrial property developers in singapore] developers in singapore and fitness. Discovers the beauty in going to places around the entire world, of late just coming back from San Marino Historic Centre and Mount Titano. |
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| ==Definitions==
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| A connected dominating set of a graph ''G'' is a set ''D'' of vertices with two properties:
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| #Any node in ''D'' can reach any other node in ''D'' by a path that stays entirely within ''D''. That is, ''D'' [[induced subgraph|induces]] a connected subgraph of ''G''.
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| #Every vertex in ''G'' either belongs to ''D'' or is adjacent to a vertex in ''D''. That is, ''D'' is a [[dominating set]] of ''G''.
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| A '''minimum connected dominating set''' of a graph ''G'' is a connecting dominating set with the smallest possible [[cardinality]] among all connected dominating sets of ''G''. The '''connected domination number''' of ''G'' is the number of vertices in the minimum connected dominating set.<ref>{{citation
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| | last1 = Sampathkumar | first1 = E.
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| | last2 = Walikar | first2 = HB
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| | issue = 6
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| | journal = J. Math. Phys. Sci
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| | pages = 607–613
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| | title = The connected domination number of a graph
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| | volume = 13
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| | year = 1979}}.</ref>
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| Any [[spanning tree]] ''T'' of a graph ''G'' has at least two leaves, vertices that have only one edge of ''T'' incident to them. A maximum leaf spanning tree is a spanning tree that has the largest possible number of leaves among all spanning trees of ''G''. The '''max leaf number''' of ''G'' is the number of leaves in the maximum leaf spanning tree.<ref name="ecology">{{citation
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| | last1 = Fellows | first1 = Michael
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| | last2 = Lokshtanov | first2 = Daniel
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| | last3 = Misra | first3 = Neeldhara
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| | last4 = Mnich | first4 = Matthias
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| | last5 = Rosamond | first5 = Frances
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| | last6 = Saurabh | first6 = Saket
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| | doi = 10.1007/s00224-009-9167-9
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| | issue = 4
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| | journal = Theory of Computing Systems
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| | pages = 822–848
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| | title = The complexity ecology of parameters: an illustration using bounded max leaf number
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| | volume = 45
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| | year = 2009}}.</ref>
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| ==Complementarity==
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| If ''d'' is the connected domination number of an ''n''-vertex graph ''G'', and ''l'' is its max leaf number, then the three quantities ''d'', ''l'', and ''n'' obey the simple equation
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| :<math>\displaystyle n = d + l.</math><ref name="d92">{{citation
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| | last = Douglas | first = Robert J.
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| | doi = 10.1016/0012-365X(92)90130-8
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| | issue = 1–3
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| | journal = Discrete Mathematics
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| | pages = 41–47
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| | title = NP-completeness and degree restricted spanning trees
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| | volume = 105
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| | year = 1992}}.</ref>
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| If ''D'' is a connected dominating set, then there exists a [[spanning tree]] in ''G'' whose leaves include all vertices that are not in ''D'': form a spanning tree of the subgraph induced by ''D'', together with edges connecting each remaining vertex ''v'' that is not in ''D'' to a neighbor of ''v'' in ''D''. This shows that {{nowrap|''l'' ≥ ''n'' − ''d''.}}
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| In the other direction, if ''T'' is any spanning tree in ''G'', then the vertices of ''T'' that are not leaves form a connected dominating set of ''G''. This shows that {{nowrap|''n'' − ''l'' ≥ ''d''.}} Putting these two inequalities together proves the equality {{nowrap|1=''n'' = ''d'' + ''l''.}}
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| Therefore, in any graph, the sum of the connected domination number and the max leaf number equals the total number of vertices.
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| Computationally, this implies that finding the minimum dominating set is equally difficult to finding a maximum leaf spanning tree.
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| ==Algorithms==
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| It is [[NP-complete]] to test whether there exists a connected dominating set with size less than a given threshold, or equivalently to test whether there exists a spanning tree with at least a given number of leaves. Therefore, it is believed that the minimum connected dominating set problem and the maximum leaf spanning tree problem cannot be solved in polynomial time.
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| When viewed in terms of approximation algorithms, connected domination and maximum leaf spanning trees are not the same: approximating one to within a given [[approximation ratio]] is not the same as approximating the other to the same ratio.
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| There exists an approximation for the minimum connected dominating set that achieves a factor of {{nowrap|2 ln Δ + O(1)}}, where Δ is the maximum degree of a vertex in G.<ref>{{citation
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| | last1 = Guha | first1 = S.
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| | last2 = Khuller | first2 = S.
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| | doi = 10.1007/PL00009201
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| | issue = 4
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| | journal = Algorithmica
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| | pages = 374–387
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| | title = Approximation algorithms for connected dominating sets
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| | volume = 20
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| | year = 1998}}.</ref>
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| The maximum leaf spanning tree problem is [[MAX-SNP]] hard, implying that no [[polynomial time approximation scheme]] is likely.<ref>{{citation
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| | last1 = Galbiati | first1 = G.
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| | last2 = Maffioli | first2 = F.
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| | last3 = Morzenti | first3 = A.
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| | doi = 10.1016/0020-0190(94)90139-2
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| | issue = 1
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| | journal = Information Processing Letters
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| | pages = 45–49
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| | title = A short note on the approximability of the maximum leaves spanning tree problem
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| | volume = 52
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| | year = 1994}}.</ref> However, it can be approximated to within a factor of 2 in polynomial time.<ref>{{citation
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| | last = Solis-Oba | first = Roberto
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| | contribution = 2-approximation algorithm for finding a spanning tree with maximum number of leaves
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| | doi = 10.1007/3-540-68530-8_37
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| | pages = 441–452
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Computer Science
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| | title = [[European Symposium on Algorithms|Proc. 6th European Symposium on Algorithms (ESA'98)]]
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| | volume = 1461
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| | year = 1998}}.</ref>
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| ==Applications==
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| Connected dominating set are useful in the computation of [[routing]] for [[mobile ad hoc network]]s. In this application, a small connected dominating set is used as a backbone for communications, and nodes that are not in this set communicate by passing messages through neighbors that are in the set.<ref>{{citation
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| | last1 = Wu | first1 = J.
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| | last2 = Li | first2 = H.
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| | contribution = On calculating connected dominating set for efficient routing in ad hoc wireless networks
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| | doi = 10.1145/313239.313261
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| | pages = 7–14
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| | publisher = ACM
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| | title = Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications
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| | year = 1999}}.</ref>
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| The max leaf number has been employed in the development of [[fixed-parameter tractability|fixed-parameter tractable]] [[algorithm]]s: several NP-hard optimization problems may be solved in polynomial time for graphs of bounded max leaf number.<ref name="ecology"/>
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| ==References==
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| {{reflist}}
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| [[Category:Computational problems in graph theory]]
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| [[Category:Graph connectivity]]
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38 year-old Electronic Device Deals Staff (Standard ) Cameron Kull from Timmins, likes r/c cars, industrial property developers in singapore developers in singapore and fitness. Discovers the beauty in going to places around the entire world, of late just coming back from San Marino Historic Centre and Mount Titano.