### Abstract

In a graph G whose vertices contain pebbles, a pebbling move uv removes two pebbles from u and adds one pebble to a neighbor v of u. The optimal pebbling number π̂(G) is the minimum k such that there exists a distribution of k pebbles to G so that for any target vertex r in G, there is a sequence of pebbling moves which places a pebble on r. The pebbling number π(G) is the minimum k such that for all distributions of k pebbles to G and for any target vertex r, there is a sequence of pebbling moves which places a pebble on r. We explore the computational complexity of computing π̂(G) and π(G). In particular, we show that deciding whether π̂(G) ≤ k is NP-complete. Furthermore, we prove that deciding whether π(G) ≤ k is ∏_{2}^{P}-complete and therefore both NP-hard and coNP-hard. Additionally, we provide a characterization of when an unordered set of pebbling moves can be ordered to form a valid sequence of pebbling moves.

Original language | English (US) |
---|---|

Pages (from-to) | 769-798 |

Number of pages | 30 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 20 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2006 |

### Fingerprint

### Keywords

- Complexity
- Graph pebbling
- ∏-completeness

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*20*(3), 769-798. https://doi.org/10.1137/050636218

**The complexity of graph pebbling.** / Milans, Kevin; Clark, Bryan.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 20, no. 3, pp. 769-798. https://doi.org/10.1137/050636218

}

TY - JOUR

T1 - The complexity of graph pebbling

AU - Milans, Kevin

AU - Clark, Bryan

PY - 2006/12/1

Y1 - 2006/12/1

N2 - In a graph G whose vertices contain pebbles, a pebbling move uv removes two pebbles from u and adds one pebble to a neighbor v of u. The optimal pebbling number π̂(G) is the minimum k such that there exists a distribution of k pebbles to G so that for any target vertex r in G, there is a sequence of pebbling moves which places a pebble on r. The pebbling number π(G) is the minimum k such that for all distributions of k pebbles to G and for any target vertex r, there is a sequence of pebbling moves which places a pebble on r. We explore the computational complexity of computing π̂(G) and π(G). In particular, we show that deciding whether π̂(G) ≤ k is NP-complete. Furthermore, we prove that deciding whether π(G) ≤ k is ∏2P-complete and therefore both NP-hard and coNP-hard. Additionally, we provide a characterization of when an unordered set of pebbling moves can be ordered to form a valid sequence of pebbling moves.

AB - In a graph G whose vertices contain pebbles, a pebbling move uv removes two pebbles from u and adds one pebble to a neighbor v of u. The optimal pebbling number π̂(G) is the minimum k such that there exists a distribution of k pebbles to G so that for any target vertex r in G, there is a sequence of pebbling moves which places a pebble on r. The pebbling number π(G) is the minimum k such that for all distributions of k pebbles to G and for any target vertex r, there is a sequence of pebbling moves which places a pebble on r. We explore the computational complexity of computing π̂(G) and π(G). In particular, we show that deciding whether π̂(G) ≤ k is NP-complete. Furthermore, we prove that deciding whether π(G) ≤ k is ∏2P-complete and therefore both NP-hard and coNP-hard. Additionally, we provide a characterization of when an unordered set of pebbling moves can be ordered to form a valid sequence of pebbling moves.

KW - Complexity

KW - Graph pebbling

KW - ∏-completeness

UR - http://www.scopus.com/inward/record.url?scp=34547863895&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547863895&partnerID=8YFLogxK

U2 - 10.1137/050636218

DO - 10.1137/050636218

M3 - Article

AN - SCOPUS:34547863895

VL - 20

SP - 769

EP - 798

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -