Graph pebbling

Graph pebbling is a mathematical game 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.[1]

π(G) for families of graphs

The pebbling number is known for the following families of graphs:

$\pi (K_{n})\,=\,n$ , where $K_{n}$ is a complete graph on n vertices.[1]
$\pi (P_{n})\,=\,2^{n-1}$ , where $P_{n}$ is a path graph on n vertices.[1]
$\pi (W_{n})\,=\,n$ , where $W_{n}$ is a wheel graph on n vertices.

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

Crull et al. 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.[2] A result called the stacking theorem finds the cover pebbling number for any graph.[3][4]

The stacking theorem

According to the stacking theorem, the initial configuration of pebbles that requires the most pebbles to be cover solved happens when all pebbles are placed on a single vertex. Based on this observation, define

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

for every vertex v in G, where 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

The cover pebbling number is known for the following families of graphs:

$\gamma (K_{n})\,=\,2n-1$ , where $K_{n}$ is a complete graph on n vertices.
$\gamma (P_{n})\,=\,2^{n}-1$ , where $P_{n}$ is a path on n vertices.
$\gamma (W_{n})\,=\,4n-9$ , where $W_{n}$ is a wheel graph on n vertices.[5]

References

Chung, Fan R. K. (1989). "Pebbling in hypercubes". SIAM Journal on Discrete Mathematics. 2 (4): 467–472. doi:10.1137/0402041. MR 1018531.CS1 maint: ref=harv (link)
Crull, Betsy; Cundiff, Tammy; Feltman, Paul; Hurlbert, Glenn H.; Pudwell, Lara; Szaniszlo, Zsuzsanna; Tuza, Zsolt (2005), "The cover pebbling number of graphs" (PDF), Discrete Mathematics, 296 (1): 15–23, doi:10.1016/j.disc.2005.03.009, MR 2148478
Vuong, Annalies; Wyckoff, M. Ian (October 18, 2004). "Conditions for Weighted Cover Pebbling of Graphs". arXiv:math/0410410.
Sjöstrand, Jonas (2005). "The cover pebbling theorem". Electronic Journal of Combinatorics. 12: Note 22. MR 2180807.
Watson, Nathaniel G.; Yerger, Carl R. (2006). "Cover pebbling numbers and bounds for certain families of graphs". Bulletin of the Institute of Combinatorics and its Applications. 48: 53–62. arXiv:math/0409321. MR 2259702.