Graph pebbling

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 (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:

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.

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.

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 the literature^[1] 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.

π(G) for families of graphs

$\scriptstyle\pi(K_n)\, =\, n$ where $K_n$ is a complete graph on n vertices.

$\scriptstyle\pi(P_n)\, =\, 2^{n-1}$ where $P_n$ is a path graph on n vertices.

$\scriptstyle\pi(W_n)\, =\, n$ where $W_n$ is a wheel graph on n vertices.

γ(G) — the cover pebbling number of a graph

Crull et al.^[2] 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^[3] 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.^[4]

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:

s(v) = \sum_{u \in V(G)} 2^{d(u,v)}

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.

γ(G) for families of graphs

$\scriptstyle \gamma(K_n)\, =\, 2n - 1$ where $\scriptstyle K_n$ is a complete graph on n vertices.

$\scriptstyle\gamma(P_n)\, =\, 2^{n}-1$ where $\scriptstyle P_n$ is a path on n vertices.

$\scriptstyle \gamma (W_n)\, =\, 4n - 9$ where $\scriptstyle W_n$ is a wheel graph on n vertices.

References

↑ F.R.K. Chung (1989). "Pebbling in Hypercubes". SIAM Journal on Discrete Mathematics 2 (4): 467–472. doi:10.1137/0402041.
↑ Betsy Crull; Tammy Cundiff; Paul Feltman; Glenn Hurlbert; Lara Pudwell; Zsuzsanna Szaniszlo; Zsolt Tuza (2005). "The cover pebbling number of graphs" (PDF). Discrete Math. 296: 15–23. doi:10.1016/j.disc.2005.03.009.
↑ Annalies Vuong; M. Ian Wyckoff (18 October 2004). "Conditions for Weighted Cover Pebbling of Graphs". arXiv:math/0410410 [math.CO].
↑ Sjöstrand, Jonas (2005). "The Cover Pebbling Theorem". Electronic Journal of Combinatorics 12 (1): N12. Retrieved 2008-08-02.

Glenn Hurlbert (1999). "A survey of graph pebbling" (PDF). Congressus Numerantium 139: 41–64.
Lior Pachter; Hunter Snevily and Bill Voxman (1994). "On Pebbling Graphs" (PDF). Congressus Numerantium 107: 65–80.