# Graph pebbling

## From Wikipedia, the free encyclopedia

**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

npebbles 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:

- , where is a complete graph on
*n*vertices.^{[1]} - , where is a path graph on
*n*vertices.^{[1]} - , where 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

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:

- , where is a complete graph on
*n*vertices. - , where is a path on
*n*vertices. - , where is a wheel graph on
*n*vertices.^{[5]}

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