1. Construction We constructed the following graph: This graph has several nice properties: Diameter Two Graph Pebbling Tim Lewis 1, Dan Simpson 1, Sam Taggart 2, and Dr. Charles Cusack 1 For more information contact: Dr. Charles Cusack 233 VanderWerf Hall 27 Graves Place Holland, MI 49423 cusack@hope.edu Phone: (616) 395-7271 FAX: (616) 395-7123 A pebbling move. What Is Graph Pebbling? A graph is a mathematical object comprised of vertices and edges connecting certain pairs of vertices. Graph pebbling is a mathematical Given a graph and a pebbling configuration, we can ask a few questions: The distance between vertices u and v is measured by counting the number of edges in the shortest path between u and v. The diameter of a graph is the greatest distance between any two of its vertices. A diameter two graph. game played on a graph. Some number of pebbles are placed the vertices. One can move a pebble from a vertex to a neighbor in the graph at the cost of one additional pebble from the source vertex. Can you get one pebble to the red vertex? If so, the vertex is said to be reachable. This problem is called REACHABLE, and is NP-complete. Is every vertex in the graph reachable? If so, it’s said to be solvable. This problem is called SOLVABLE, and it’s NP- complete. How many pebbles can you place on this graph before it’s guaranteed to be solvable? This number is called the pebbling number, and finding it is much harder than NP-complete. NP-complete is the class of decision (yes- or-no) problems that are: NP: given a solution to a problem in NP, we can verify that it is a solution relatively quickly (in polynomial time). NP-hard: they are at least as “hard” as every other problem in NP. “Yes” Instances “No” Instances “Yes” Instances “No” Instances Reduction By preserving “Yes” and “No” instances, a reduction from A to B allows us to use any algorithm that solves B to solve A. Pebble Bounds Converting to Diameter Two We can take a graph and transform it into a diameter two graph without changing the reachability of any vertex. Embed the graph into the one-element vertices of the above construction. This only works if no vertex in the original graph can get more than three pebbles. The above construction fulfills these requirements and has. vertices. We were able to improve on this a little and remove vertices from the construction and prove that this is best possible. Solving for n tells us that if a diameter two graph has more than vertices with two pebbles, then it must be solvable. Acknowledgements Computer Science Department at Hope College National Science Foundation k = 4 Note the added edges and missing vertices Two Reductions Restricted 3-SAT Variable Gadget Red=False Blue=True The Punch Line: Reachability in diameter two graphs is NP-complete. Using the chain of reductions, we were able to create a reduction from R3-SAT to REACHABLE in diameter two graphs. x 1 = False x 3 = True x 2 = True It is impossible to move three pebbles onto any vertex in this construction (Try it!). This lets us use another one of our constructions to make our reduction diameter two. 1: Hope College, 2: Oberlin College graph.computinggames.org Future Work Having obtained a tight bound on the number of vertices with two pebbles in an unsolvable diameter two graph, we would like to look into what sort of implications this bound has on the random graph pebbling. We suspect that this will yield results relating to the threshold of such graphs, but there is more work to be done. In the future, we intend to devise an algorithm to calculate the pebbling number of a graph. Currently we have an exhaustive algorithm to do this, but it is useful only for small graphs due to the massive amount of computation that must be done. By skipping configurations that are already known to be solvable, we should be able to implement an algorithm that will run many times faster while still accurately calculating the pebbling number. SymbolNameFunction VariableEither True or False. ORTrue if and only if at least one input is T. ANDTrue if and only if both inputs are T. NOTSwitches the value of its input. Boolean Algebra Example: Try to find an assignment of the variables that makes this expression true. This problem is called SATISFIABLE. It’s NP-complete, even when the expressions are comprised of clauses (containing variables, ORs, and NOTs) connected by ANDs. This restricted version is called R3-SAT. NP-completeness To prove that decision problem A is “harder” than decision problem B, we use a function called a reduction. The function takes instances of A to instances of B in a very particular way: Reductions are usually required to be computable in polynomial time, so an algorithm to quickly solve one NP-complete problem could be used to solve any of the others. No such algorithm has been found, and the question of whether or not such an algorithm even exists remains one of the most important questions in mathematics and computer science (P vs. NP). Examples of NP-complete problems: Traveling Salesman Crossword Puzzle Construction Graph Coloring Sudoku Bin Packing Tetris Diameter two Three pebbles can be placed on all but one of the one-element vertices and the graph is unsolvable. k = 5 We can turn an instance of R3-SAT into a REACHABLE problem. Each piece of the boolean expression will be turned into a graph structure: Vertices are labeled with the one- and two-element subsets of {1,…,k}. Vertices that share an element are connected with an edge. adjacent vertices If we take a graph and keep adding pebbles to it, eventually you will be guaranteed that the graph is solvable. What we wanted to know was for a diameter two graph, on how many vertices can you place two pebbles before the graph is guaranteed to be solvable. This problem is the same as finding the smallest diameter two graph with k vertices with two pebbles. This has applications in random pebbling distributions and pebbling thresholds. OR Gadget Input 1Input 2Input 3 Output The OR gadget places one pebble on the output when it gets at least one input. Target AND Gadget The AND gadget can place one pebble on the target only if it receives one pebble on each input node. Inputs