Graph pebbling
From Wikipedia, the free encyclopedia
Graph pebbling is a mathematical game and area of interest played on a graph with pebbles on the 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. π(G), the pebbling number of a graph G is the lowest natural number that fulfills the following property:
Given any target or 'root' vertex in the graph and any initial configuration of π(G) 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.
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.
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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[edit] π(G) — the pebbling number of a graph
The game of pebbling was first suggested by Lagarias and Saks, as a tool for solving a particular problem in number theory. In 1989 F.R.K. Chung introduced the concept in literature and defined the pebbling number, π(G).
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.
[edit] π(G) for families of graphs
where Kn is a complete graph on n vertices.
Where Pn is a path graph on n vertices.
where Wn is a wheel graph on n vertices.
[edit] γ(G) — the cover pebbling number of a graph
Crull et al. 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 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.
[edit] The stacking theorem
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:
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.
[edit] γ(G) for families of graphs
where 
is a complete graph on n vertices.
where 
is a path graph on n vertices.
where 
is a wheel graph on n vertices.
[edit] References
- Glenn Hurlbert, A survey of graph pebbling, Congressus Numerantium 139 (1999), 41--64. Available via pdf at [1]
- Lior Pachter, Hunter Snevily and Bill Voxman, "On Pebbling Graphs", Congressus Numerantium 107 (1994), 65--80. Available via pdf at [2]
- Betsy Crull, Tammy Cundiff, Paul Feltman, Glenn Hurlbert, Lara Pudwell, Zsuzsanna Szaniszlo, Zsolt Tuza, The cover pebbling number of graphs, Discrete Math. 296 (2005), 15--23. vailable via pdf at [3]
