68 research outputs found
A linear optimization technique for graph pebbling
Graph pebbling is a network model for studying whether or not a given supply
of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling
move across an edge of a graph takes two pebbles from one endpoint and places
one pebble at the other endpoint; the other pebble is lost in transit as a
toll. It has been shown that deciding whether a supply can meet a demand on a
graph is NP-complete. The pebbling number of a graph is the smallest t such
that every supply of t pebbles can satisfy every demand of one pebble. Deciding
if the pebbling number is at most k is \Pi_2^P-complete. In this paper we
develop a tool, called the Weight Function Lemma, for computing upper bounds
and sometimes exact values for pebbling numbers with the assistance of linear
optimization. With this tool we are able to calculate the pebbling numbers of
much larger graphs than in previous algorithms, and much more quickly as well.
We also obtain results for many families of graphs, in many cases by hand, with
much simpler and remarkably shorter proofs than given in previously existing
arguments (certificates typically of size at most the number of vertices times
the maximum degree), especially for highly symmetric graphs. Here we apply the
Weight Function Lemma to several specific graphs, including the Petersen,
Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a
number of infinite families of graphs, such as trees, cycles, graph powers of
cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly
answers a question of Pachter, et al., by computing the pebbling exponent of
cycles to within an asymptotically small range. It is conceivable that this
method yields an approximation algorithm for graph pebbling
Optimal Pebbling Number of the Square Grid
A pebbling move on a graph removes two pebbles from a vertex and adds one
pebble to an adjacent vertex. A vertex is reachable from a pebble distribution
if it is possible to move a pebble to that vertex using pebbling moves. The
optimal pebbling number is the smallest number m needed to
guarantee a pebble distribution of m pebbles from which any vertex is
reachable. The optimal pebbling number of the square grid graph was investigated in several papers. In this paper, we present a new method
using some recent ideas to give a lower bound on . We apply this
technique to prove that . Our
method also gives a new proof for
The optimal pebbling number of staircase graphs
Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can be placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs. © 2018 Elsevier B.V
Roman Domination Cover Rubbling
In this thesis, we introduce Roman domination cover rubbling as an extension of domination cover rubbling. We define a parameter on a graph called the \textit{Roman domination cover rubbling number}, denoted , as the smallest number of pebbles, so that from any initial configuration of those pebbles on , it is possible to obtain a configuration which is Roman dominating after some sequence of pebbling and rubbling moves. We begin by characterizing graphs having small value. Among other things, we also obtain the Roman domination cover rubbling number for paths and give an upper bound for the Roman domination cover rubbling number of a tree
A linear optimization technique for graph pebbling
Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble. Deciding if the pebbling number is at most k is NP 2 -complete. In this paper we develop a tool, called theWeight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply theWeight Function Lemma to several specific graphs, including the Petersen, Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly answers a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling
The pebbling threshold of the square of cliques
AbstractGiven an initial configuration of pebbles on a graph, one can move pebbles in pairs along edges, at the cost of one of the pebbles moved, with the objective of reaching a specified target vertex. The pebbling number of a graph is the minimum number of pebbles so that every configuration of that many pebbles can reach any chosen target. The pebbling threshold of a sequence of graphs is roughly the number of pebbles so that almost every (resp. almost no) configuration of asymptotically more (resp. fewer) pebbles can reach any chosen target. In this paper we find the pebbling threshold of the sequence of squares of cliques, improving upon an earlier result of Boyle and verifying an important instance of a probabilistic version of Graham's product conjecture
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