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| '''Graph pebbling''' is a mathematical [[game]] and area of interest played on a [[graph (mathematics)|graph]] with pebbles on the [[vertex (graph theory)|vertices]]. 'Game play' is composed of a series of pebbling moves. A pebbling move on a graph consists of taking two pebbles off one vertex and placing one on an adjacent vertex (the second removed pebble is discarded from play). π(''G''), the pebbling number of a graph ''G'' is the lowest [[natural number]] ''n'' that satisfies the following condition:
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| <blockquote>
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| Given any target or 'root' vertex in the graph and any initial configuration of ''n'' pebbles on the graph, it is possible, after a series of pebbling moves, to reach a new configuration in which the designated root vertex has one or more pebbles.
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| </blockquote>
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| For example, on a graph with 2 vertices and 1 edge connecting them the pebbling number is 2. No matter how the two pebbles are placed on the vertices of the graph it is always possible to move a pebble to any vertex in the graph. One of the central questions of graph pebbling is the value of π(''G'') for a given graph ''G''.
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| Other topics in pebbling include cover pebbling, optimal pebbling, domination cover pebbling, bounds, and thresholds for pebbling numbers, deep graphs, and others.
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| ==π(''G'') — the pebbling number of a graph==
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| The game of pebbling was first suggested by Lagarias and Saks, as a tool for solving a particular problem in [[number theory]]. In 1989 [[Fan Chung|F.R.K. Chung]] introduced the concept in the literature<ref>{{cite journal | author=F.R.K. Chung | title=Pebbling in Hypercubes | journal=[[SIAM Journal on Discrete Mathematics]] | volume=2 | year=1989 | pages=467–472 | doi=10.1137/0402041 | issue=4 }}</ref> and defined the pebbling number, π(''G'').
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| The pebbling number for a [[complete graph]] on ''n'' vertices is easily verified to be ''n'': If we had (''n'' − 1) pebbles to put on the graph, then we could put 1 pebble on each vertex except one. This would make it impossible to place a pebble on the last vertex. Since this last vertex could have been the designated target vertex, the pebbling number must be greater than ''n'' − 1. If we were to add 1 more pebble to the graph there are 2 possible cases. First, we could add it to the empty vertex, which would put a pebble on every vertex. Or secondly, we could add it to one of the vertices with only 1 pebble on them. Once any vertex has 2 pebbles on it, it becomes possible to make a pebbling move to any other vertex in the complete graph.
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| ===π(''G'') for families of graphs===
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| <math>\scriptstyle\pi(K_n)\, =\, n</math> where <math>K_n</math> is a [[complete graph]] on ''n'' vertices.
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| <math>\scriptstyle\pi(P_n)\, =\, 2^{n-1}</math> where <math>P_n</math> is a [[Path (graph theory)|path graph]] on ''n'' vertices.
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| <math>\scriptstyle\pi(W_n)\, =\, n</math> where <math>W_n</math> is a [[wheel graph]] on ''n'' vertices.
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| ==γ(''G'') — the cover pebbling number of a graph==
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| Crull et al.<ref>{{cite journal | author=Betsy Crull | coauthors=Tammy Cundiff, Paul Feltman, Glenn Hurlbert, Lara Pudwell, Zsuzsanna Szaniszlo, Zsolt Tuza | title=The cover pebbling number of graphs | journal=Discrete Math. | volume=296 | year=2005 | pages=15–23 | format=pdf | url=http://mingus.la.asu.edu/~hurlbert/pebbling/papers/CCFHPSZ_CPN.pdf | doi=10.1016/j.disc.2005.03.009 }}</ref> introduced the concept of cover pebbling. γ(''G''), the cover pebbling number of a graph is the minimum number of pebbles needed so that from any initial arrangement of the pebbles, after a series of pebbling moves, it is possible to have at least 1 pebble on every vertex of a graph. Vuong and Wyckoff<ref>{{cite arxiv | title=Conditions for Weighted Cover Pebbling of Graphs | author=Annalies Vuong | coauthors=M. Ian Wyckoff | eprint=math/0410410 | date=18 October 2004 | class=math.CO}}</ref> proved a theorem known as the stacking theorem which essentially finds the cover pebbling number for any graph: this theorem was proved at about the same time by Jonas Sjostrand.<ref>{{cite journal | author=Sjöstrand, Jonas | title=The Cover Pebbling Theorem | journal=Electronic Journal of Combinatorics | volume=12 | issue=1 | pages=N12 | year=2005 | url=http://www.combinatorics.org/Volume_12/Abstracts/v12i1n22.html | accessdate=2008-08-02}}</ref>
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| ===The stacking theorem===
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| The stacking theorem states the initial configuration of pebbles that requires the most pebbles to be cover solved happens when all pebbles are placed on a single vertex. From there they state:
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| : <math>s(v) = \sum_{u \in V(G)} 2^{d(u,v)}</math>
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| Do this for every vertex ''v'' in ''G''. ''d''(''u'', ''v'') denotes the distance from ''u'' to ''v''. Then the cover pebbling number is the largest ''s''(''v'') that results.
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| ===γ(''G'') for families of graphs===
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| <math>\scriptstyle \gamma(K_n)\, =\, 2n - 1</math> where <math>\scriptstyle K_n</math> is a [[complete graph]] on ''n'' vertices.
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| <math>\scriptstyle\gamma(P_n)\, =\, 2^{n}-1</math> where <math>\scriptstyle P_n</math> is a [[Path (graph theory)|path]] on ''n'' vertices.
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| <math>\scriptstyle \gamma (W_n)\, =\, 4n - 5</math> where <math>\scriptstyle W_n</math> is a [[wheel graph]] on ''n'' vertices.
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * {{cite journal | author=Glenn Hurlbert | title=A survey of graph pebbling | journal=Congressus Numerantium | volume=139 | year=1999 | pages=41–64 | format=pdf | url=http://mingus.la.asu.edu/~hurlbert/pebbling/papers/Hurl_SGP.pdf }}
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| * {{cite journal | author=Lior Pachter | coauthors=Hunter Snevily and Bill Voxman | title=On Pebbling Graphs | journal=Congressus Numerantium | volume=107 | year=1994 | pages=65–80 | format=pdf | url=http://math.berkeley.edu/~lpachter/papers/pebble.pdf }}
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| {{refend}}
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| ==External links==
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| {{DEFAULTSORT:Graph Pebbling}}
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| [[Category:Games of mental skill]]
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| [[Category:Mathematical games]]
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