Abstract

Given a simple, connected graph, a pebbling configuration is a function from its vertex set to the nonnegative integers. A pebbling move between adjacent vertices removes two pebbles from one vertex and adds one pebble to the other. A vertex $r$ is said to be reachable from a configuration if there exists a sequence of pebbling moves that places one pebble on $r$. A configuration is solvable if every vertex is reachable. We prove tight bounds on the number of vertices with two and three pebbles that an unsolvable configuration on a diameter two graph can have in terms of the size of the graph. We also prove that determining reachability of a vertex is NP-complete, even in graphs of diameter two.

Keywords

  1. graph pebbling
  2. diameter two
  3. NP-complete

MSC codes

  1. 68Q17
  2. 68R05
  3. 68R10

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
A. Bekmetjev, G. Brightwell, A. Czygrinow, and G. Hurlbert, Thresholds for families of multisets, with an application to graph pebbling, Discrete Math., 269 (2003), pp. 21--34.
2.
A. Bekmetjev and C. A. Cusack, Pebbling algorithms in diameter two graphs, SIAM J. Discrete Math., 23 (2009), pp. 634--646.
3.
A. Blasiak and J. Schmitt, Degree sum conditions in graph pebbling, Australas. J. Combin., 42 (2008), pp. 83--90.
4.
F. R. K. Chung, Pebbling in hypercubes, SIAM J. Discrete Math., 2 (1989), pp. 467--472.
5.
T. A. Clarke, R. A. Hochberg, and G. H. Hurlbert, Pebbling in diameter two graphs and products of graphs, J. Graph Theory, 25 (1997), pp. 119--128.
6.
D. Herscovici, B. Hester, and G. Hurlbert, t-pebbling and extensions, Graphs Combin., March 2012 (electronic).
7.
G. Hurlbert, A survey of graph pebbling, Congr. Numer., 139 (1999), pp. 41--64.
8.
G. Hurlbert, Recent progress in graph pebbling, Graph Theory Notes N.Y., XLIX (2005), pp. 25--37.
9.
G. Hurlbert and H. Kierstead, On the Complexity of Graph Pebbling, unpublished manuscript, School of Mathematical and Statistical Sciences, Arizona State University, 2005.
10.
K. Milans and B. Clark, The complexity of graph pebbling, SIAM J. Discrete Math., 20 (2006), pp. 769--798.
11.
N. G. Watson, The Complexity of Pebbling and Cover Pebbling, preprint, 2005, online at arXiv:math/0503511v3.

Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 919 - 928
ISSN (online): 1095-7146

History

Submitted: 11 August 2011
Accepted: 6 April 2012
Published online: 10 July 2012

Keywords

  1. graph pebbling
  2. diameter two
  3. NP-complete

MSC codes

  1. 68Q17
  2. 68R05
  3. 68R10

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media