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.

Given a configuration of t indistinguishable pebbles on the n vertices of a graph G, we say that a vertex v can be reached if a pebble can be placed on it in a finite number of "moves". G is said to be pebbleable if all its vertices can be thus reached. Now given the n-path Pn how large (resp. small) must t be so as to be able to pebble the path almost surely (resp. almost never)? It was known that the threshold th(Pn) for pebbling the path satisfies n2clgn≤th(Pn)≤n22lgn, where lg=log2 and c<1/2 is arbitrary. We improve the upper bound for the threshold function to th(Pn)≤n2dlgn, where d>1 is arbitrary.

For a graph G=(V,E), we consider placing a variable number of pebbles on the vertices of V. A pebbling move consists of deleting two pebbles from a vertex u∈V and placing one pebble on a vertex v adjacent to u. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than some positive integer t pebbles and for any given vertex v∈V, it is possible, by a sequence of pebbling moves, to move at least one pebble to v. We relate this minimum number of pebbles to several other well-studied parameters of a graph G, including the domination number, the optimal pebbling number, and the Roman domination number of G.

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.

Currently, the most effective complete SAT solvers are based on the DPLL algorithm augmented by Clause Learning. These solvers can handle many real-world problems from application areas like verification, diagnosis, planning, and design. Clause Learning works by storing previously computed, intermediate results and then reusing them to prune the future search tree. Without Clause Learning, however, DPLL loses most of its effectiveness on real world problems. Recently there has been some work on obtaining a deeper understanding of the technique of Clause Learning. In this thesis, we contribute to the understanding of Clause Learning, and the Resolution proof system that underlies it, in a number of ways. We characterize Clause Learning as a new, intuitive Resolution refinement which we call CL. We then show that CL proofs can effectively p-simulate general Resolution. Furthermore, this result holds even for the more restrictive class of greedy, unit propagating CL proofs, which more accurately characterize Clause Learning as it is used in practice. This result is surprising and indicates that Clause Learning is significantly more powerful than was previously known. Since Clause Learning makes use of previously derived clauses, it motivates the study of Resolution space. We contribute to this area of study by proving that determining the variable space of a Resolution derivation is PSPACE-complete. The reduction also yields a surprising exponential size/space trade-off for Resolution in which an increase of just 3 units of variable space results in an exponential decrease in proofsize. This result runs counter to the intuitions of many in the SAT-solving community who have generally believed that proof-size should decrease smoothly as available space increases. In order to prove these Resolution results, we need to make use of some intuition regarding the relationship between Black-White Pebbling and Resolution. In fact, determining the complexity of Resolution variable space required us to first prove that Black-White Pebbling is PSPACE-complete. The complexity of the Black-White Pebbling Game has remained an open problem for 30 years and resisted numerous attempts to solve it. Its solution is the primary contribution of this thesis.

Currently, the most effective complete SAT solvers are based on the DPLL algorithm augmented by Clause Learning. These solvers can handle many real-world problems from application areas like verification, diagnosis, planning, and design. Clause Learning works by storing previously computed, intermediate results and then reusing them to prune the future search tree. Without Clause Learning, however, DPLL loses most of its effectiveness on real world problems. Recently there has been some work on obtaining a deeper understanding of the technique of Clause Learning. In this thesis, we contribute to the understanding of Clause Learning, and the Resolution proof system that underlies it, in a number of ways. We characterize Clause Learning as a new, intuitive Resolution refinement which we call CL. We then show that CL proofs can effectively p-simulate general Resolution. Furthermore, this result holds even for the more restrictive class of greedy, unit propagating CL proofs, which more accurately characterize Clause Learning as it is used in practice. This result is surprising and indicates that Clause Learning is significantly more powerful than was previously known. Since Clause Learning makes use of previously derived clauses, it motivates the study of Resolution space. We contribute to this area of study by proving that determining the variable space of a Resolution derivation is PSPACE-complete. The reduction also yields a surprising exponential size/space trade-off for Resolution in which an increase of just 3 units of variable space results in an exponential decrease in proofsize. This result runs counter to the intuitions of many in the SAT-solving community who have generally believed that proof-size should decrease smoothly as available space increases. In order to prove these Resolution results, we need to make use of some intuition regarding the relationship between Black-White Pebbling and Resolution. In fact, determining the complexity of Resolution variable space required us to first prove that Black-White Pebbling is PSPACE-complete. The complexity of the Black-White Pebbling Game has remained an open problem for 30 years and resisted numerous attempts to solve it. Its solution is the primary contribution of this thesis.

A pebble distribution places a nonnegative number of pebbles on the vertices of a graph G. In graph rubbling, the pebbles can be redistributed using pebbling and rubbling moves, typically with the goal of reaching some target pebble distribution. In graph pebbling, only the pebbling move is allowed. The cover pebbling number is the smallest k such that from any initial distribution of k pebbles, it is possible that after a series of pebbling moves there is at least one pebble on every vertex of G. The Cover Pebbling Theorem asserts that to determine the cover pebbling number of a graph, it is sufficient to consider the pebbling distributions that initially place all pebbles on a single vertex. In this paper, we prove a rubbling analogue of the Cover Pebbling Theorem, providing an answer to an open question of Belford and Sieben (2009). In addition, we prove a stronger version of the Cover Rubbling Theorem for trees.

Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted f (G), is the minimal number of pebbles such that every configuration of f (G) pebbles on G is solvable. We derive several general upper bounds on the pebbling number, improving previous results.

Let G be a connected simple graph with vertex set V and a distribution of pebbles on V. The domination cover rubbling number of G is the minimum number of pebbles, so that no matter how they are distributed, it is possible that after a sequence of pebbling and rubbling moves, the set of vertices with pebbles is a dominating set of G. We begin by characterizing the graphs having small domination cover rubbling numbers and determining the domination cover rubbling number of several common graph families. We then give a bound for the domination cover rubbling number of trees and characterize the extremal trees. Finally, we give bounds for the domination cover rubbling number of graphs in terms of their domination number and characterize a family of the graphs attaining this bound.

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.