27 research outputs found
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
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
Two-Player Graph Pebbling
Given a graph G with pebbles on the vertices, we define a pebbling move as removing two pebbles from a vertex u, placing one pebble on a neighbor v, and discarding the other pebble, like a toll. The pebbling number n(G) is the least number of pebbles needed so that every arrangement of n(G) pebbles can place a pebble on any vertex through a sequence of pebbling moves. We introduce a new variation on graph pebbling called two-player pebbling. In this, players called the mover and the defender alternate moves, with the stipulation that the defender cannot reverse the previous move. The mover wins only if they can place a pebble on a specified vertex and the defender wins if the mover cannot. We define n(G), analogously, as the minimum number of pebbles such that given every configuration of the n(G) pebbles and every specified vertex r, the mover has a winning strategy. First, we will investigate upper bounds for n(G) on various classes of graphs and find a certain structure for which the defender has a winning strategy, no matter how many pebbles are in a configuration. Then, we characterize winning configurations for both players on a special class of diameter 2 graphs. Finally, we show winning configurations for the mover on paths using a recursive argument
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
Automating Weight Function Generation in Graph Pebbling
Graph pebbling is a combinatorial game played on an undirected graph with an
initial configuration of pebbles. A pebbling move consists of removing two
pebbles from one vertex and placing one pebble on an adjacent vertex. The
pebbling number of a graph is the smallest number of pebbles necessary such
that, given any initial configuration of pebbles, at least one pebble can be
moved to a specified root vertex. Recent lines of inquiry apply computational
techniques to pebbling bound generation and improvement.
Along these lines, we present a computational framework that produces a set
of tree strategy weight functions that are capable of proving pebbling number
upper bounds on a connected graph. Our mixed-integer linear programming
approach automates the generation of large sets of such functions and provides
verifiable certificates of pebbling number upper bounds. The framework is
capable of producing verifiable pebbling bounds on any connected graph,
regardless of its structure or pebbling properties. We apply the model to the
4th weak Bruhat to prove and to the Lemke square graph to
produce a set of certificates that verify
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
Extremal Results for Peg Solitaire on Graphs
In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families