23 research outputs found
More on the -restricted optimal pebbling number
Let be a simple graph. A function is called a configuration of pebbles on the vertices of and the
weight of is which is just the total number of
pebbles assigned to vertices. A pebbling step from a vertex to one of its
neighbors reduces by two and increases by one. A pebbling
configuration is said to be solvable if for every vertex , there
exists a sequence (possibly empty) of pebbling moves that results in a pebble
on . A pebbling configuration is a -restricted pebbling configuration
(abbreviated RPC) if for all . The -restricted
optimal pebbling number is the minimum weight of a solvable RPC
on . Chellali et.al. [Discrete Appl. Math. 221 (2017) 46-53] characterized
connected graphs having small -restricted optimal pebbling numbers and
characterization of graphs with stated as an open problem.
In this paper, we solve this problem. We improve the upper bound of the
-restricted optimal pebbling number of trees of order . Also, we study
-restricted optimal pebbling number of some grid graphs, corona and
neighborhood corona of two specific graphs.Comment: 12 pages, 11 figure
On t-Restricted Optimal Rubbling of Graphs
For a graph G = (V;E), a pebble distribution is defined as a mapping of the vertex set in to the integers, where each vertex begins with f(v) pebbles. A pebbling move takes two pebbles from some vertex adjacent to v and places one pebble on v. A rubbling move takes one pebble from each of two vertices that are adjacent to v and places one pebble on v. A vertex x is reachable under a pebbling distribution f if there exists some sequence of rubbling and pebbling moves that places a pebble on x. A pebbling distribution where every vertex is reachable is called a rubbling configuration. The t-restricted optimal rubbling number of G is the minimum number of pebbles required for a rubbling configuration where no vertex is initially assigned more than t pebbles. Here we present results on the 1-restricted optimal rubbling number and the 2- restricted optimal rubbling number
Extensions of Graph Pebbling
My thesis will consist of extensions to results that I proved at the 2004 East Tennessee State REU. Most of these results have to do with graph pebbling and various probabilistic extensions. Specifically, in Chapter 2 we compute the cover pebbling number for complete multipartite graphs and prove upper bounds for cover pebbling numbers for graphs of a specified diameter and order. We also prove that the cover pebbling decision problem is NP complete. In Chapters 3 and 4 we examine domination cover pebbling. In Chapter 5, we obtain structural and probabilistic results for deep graphs, and in Chapter 6 we compute cover pebbling probability thresholds for the complete graph
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
Pebbling number of polymers
Let be a simple graph. A function is called a configuration of pebbles on the vertices of and the
quantity is called the weight of which is
just the total number of pebbles assigned to vertices. A pebbling step from a
vertex to one of its neighbors reduces by two and increases
by one. A pebbling configuration is said to be solvable if for every
vertex , there exists a sequence (possibly empty) of pebbling moves that
results in a pebble on . The pebbling number equals the minimum
number such that every pebbling configuration with is solvable. Let be a connected graph constructed from pairwise
disjoint connected graphs by selecting a vertex of , a
vertex of , and identifying these two vertices. Then continue in this
manner inductively. We say that is a polymer graph, obtained by
point-attaching from monomer units . In this paper, we study the
pebbling number of some polymers.Comment: 15 pages, 9 figure
Optimal pebbling number of graphs with given minimum degree
Consider a distribution of pebbles on a connected graph . A pebbling move
removes two pebbles from a vertex and places one to an adjacent vertex. A
vertex is reachable under a pebbling distribution if it has a pebble after the
application of a sequence of pebbling moves. The optimal pebbling number
is the smallest number of pebbles which we can distribute in such a
way that each vertex is reachable. It was known that the optimal pebbling
number of any connected graph is at most , where
is the minimum degree of the graph. We strengthen this bound by showing that
equality cannot be attained and that the bound is sharp. If
then we further improve the bound to
. On the other hand, we show that for
arbitrary large diameter and any there are infinitely many graphs
whose optimal pebbling number is bigger than