1-Restricted Optimal Rubbling on Graphs
Let G be a graph with vertex set V and a distribution of pebbles on the vertices of V. A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing...
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Online Access: | https://doi.org/10.7151/dmgt.2102 |
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doaj-8356c4f794c44b2f8ae1c2bb70002fb82021-09-05T17:20:24ZengSciendoDiscussiones Mathematicae Graph Theory2083-58922019-05-0139257558810.7151/dmgt.2102dmgt.21021-Restricted Optimal Rubbling on GraphsBeeler Robert A.0Haynes Teresa W.1Murphy Kyle2Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-1700 USADepartment of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-1700 USA; Department of Mathematics, University of Johannesburg, Auckland Park, South AfricaDepartment of Mathematics, Iowa State University, Ames, IA 50011 USALet G be a graph with vertex set V and a distribution of pebbles on the vertices of V. A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing a pebble on v. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than one pebble and for any given vertex v ∈ V, it is possible, by a sequence of pebbling and rubbling moves, to move at least one pebble to v. This minimum number of pebbles is the 1-restricted optimal rubbling number. We determine the 1-restricted optimal rubbling numbers for Cartesian products. We also present bounds on the 1-restricted optimal rubbling number.https://doi.org/10.7151/dmgt.2102graph pebblinggraph rubblingoptimal rubblingt-restricted optimal pebbling05c78 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Beeler Robert A. Haynes Teresa W. Murphy Kyle |
spellingShingle |
Beeler Robert A. Haynes Teresa W. Murphy Kyle 1-Restricted Optimal Rubbling on Graphs Discussiones Mathematicae Graph Theory graph pebbling graph rubbling optimal rubbling t-restricted optimal pebbling 05c78 |
author_facet |
Beeler Robert A. Haynes Teresa W. Murphy Kyle |
author_sort |
Beeler Robert A. |
title |
1-Restricted Optimal Rubbling on Graphs |
title_short |
1-Restricted Optimal Rubbling on Graphs |
title_full |
1-Restricted Optimal Rubbling on Graphs |
title_fullStr |
1-Restricted Optimal Rubbling on Graphs |
title_full_unstemmed |
1-Restricted Optimal Rubbling on Graphs |
title_sort |
1-restricted optimal rubbling on graphs |
publisher |
Sciendo |
series |
Discussiones Mathematicae Graph Theory |
issn |
2083-5892 |
publishDate |
2019-05-01 |
description |
Let G be a graph with vertex set V and a distribution of pebbles on the vertices of V. A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing a pebble on v. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than one pebble and for any given vertex v ∈ V, it is possible, by a sequence of pebbling and rubbling moves, to move at least one pebble to v. This minimum number of pebbles is the 1-restricted optimal rubbling number. We determine the 1-restricted optimal rubbling numbers for Cartesian products. We also present bounds on the 1-restricted optimal rubbling number. |
topic |
graph pebbling graph rubbling optimal rubbling t-restricted optimal pebbling 05c78 |
url |
https://doi.org/10.7151/dmgt.2102 |
work_keys_str_mv |
AT beelerroberta 1restrictedoptimalrubblingongraphs AT haynesteresaw 1restrictedoptimalrubblingongraphs AT murphykyle 1restrictedoptimalrubblingongraphs |
_version_ |
1717786376729001984 |