121 research outputs found
Modified Linear Programming and Class 0 Bounds for Graph Pebbling
Given a configuration of pebbles on the vertices of a connected graph , a
\emph{pebbling move} removes two pebbles from some vertex and places one pebble
on an adjacent vertex. The \emph{pebbling number} of a graph is the
smallest integer such that for each vertex and each configuration of
pebbles on there is a sequence of pebbling moves that places at least
one pebble on .
First, we improve on results of Hurlbert, who introduced a linear
optimization technique for graph pebbling. In particular, we use a different
set of weight functions, based on graphs more general than trees. We apply this
new idea to some graphs from Hurlbert's paper to give improved bounds on their
pebbling numbers.
Second, we investigate the structure of Class 0 graphs with few edges. We
show that every -vertex Class 0 graph has at least
edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we
strengthen this lower bound to , which is best possible. Further, we
characterize the graphs where the bound holds with equality and extend the
argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.Comment: 19 pages, 8 figure
Critical Pebbling Numbers of Graphs
We define three new pebbling parameters of a connected graph , the -,
-, and -critical pebbling numbers. Together with the pebbling number, the
optimal pebbling number, the number of vertices and the diameter of the
graph, this yields 7 graph parameters. We determine the relationships between
these parameters. We investigate properties of the -critical pebbling
number, and distinguish between greedy graphs, thrifty graphs, and graphs for
which the -critical pebbling number is .Comment: 26 page
Pebbling in Semi-2-Trees
Graph pebbling is a network model for transporting discrete resources that
are consumed in transit. Deciding whether a given configuration on a particular
graph can reach a specified target is -complete, even for diameter
two graphs, and deciding whether the pebbling number has a prescribed upper
bound is -complete. Recently we proved that the pebbling number
of a split graph can be computed in polynomial time. This paper advances the
program of finding other polynomial classes, moving away from the large tree
width, small diameter case (such as split graphs) to small tree width, large
diameter, continuing an investigation on the important subfamily of chordal
graphs called -trees. In particular, we provide a formula, that can be
calculated in polynomial time, for the pebbling number of any semi-2-tree,
falling shy of the result for the full class of 2-trees.Comment: Revised numerous arguments for clarity and added technical lemmas to
support proof of main theorem bette
Domination Cover Pebbling: Structural Results
This paper continues the results of "Domination Cover Pebbling: Graph
Families." An almost sharp bound for the domination cover pebbling (DCP) number
for graphs G with specified diameter has been computed. For graphs of diameter
two, a bound for the ratio between the cover pebbling number of G and the DCP
number of G has been computed. A variant of domination cover pebbling, called
subversion DCP is introducted, and preliminary results are discussed.Comment: 15 page
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