Pebbling in Split Graphs

Discrete Applied Mathematics, 2014

Given a simple, connected graph, a pebbling configuration is a function from its vertex set to the nonnegative integers. A pebbling move between adjacent vertices removes two pebbles from one vertex and adds one pebble to the other. A vertex r is said to be reachable from a configuration if there exists a sequence of pebbling moves that places at least one pebble on r. A configuration is solvable if every vertex is reachable. We prove that determining reachability of a vertex and solvability of a configuration are NP-complete on planar graphs. We also prove that both reachability and solvability can be determined in O(n 6 ) time on planar graphs with diameter two. Finally, for outerplanar graphs, we present a linear algorithm for determining reachability and a quadratic algorithm for determining solvability. To prove this result, we provide linear algorithms to determine all possible maximal configurations of pebbles that can be placed on the endpoints of a path and on two adjacent vertices in a cycle.

2004

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n>=9 vertices having minimum degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336 vertices in each part having minimum degree at least floor(m/2)+1. Both bounds are best possible. In addition, we prove that the pebbling threshold of graphs with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when d is proportional to n.

2008

Let G be a connected graph with the vertex set V and the edge set E, where |V | = n and |E| = m. Define a pebbling configuration as a function C : V → Z+ where C(v) represents the number of pebbles placed on vertex v. For any vertex v such that C(v) ≥ 2 a pebbling step consists of placing a pebble on one of the vertices adjacent to v and discarding two pebbles from v. A configuration is called r-solvable if there is a sequence of pebbling steps that places at least one pebble on vertex r. Any such sequence is called an r-solution. A configuration is called solvable if it is r-solvable for any r ∈ V. We call an r-solution minimal if it contains the smallest number of pebbling steps. The pebbling number of a graph G, denoted π(G), is the minimum number of pebbles such that the configuration is solvable no matter how the pebbles are distriibuted on the vertices. For any two vertices u, v ∈ V , the distance between u and v (denoted d(u, v)) is the the number of edges on the shortest pat...

2011

Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble. Deciding if the pebbling number is at most k is \Pi_2^P-complete. In this paper we develop a tool, called the Weight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply the Weight Function Lemma to several specific graphs, including the Petersen, Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly answers a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling.

Discrete Applied Mathematics

Consider a distribution of pebbles on a connected graph G. 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 πopt(G) 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 4n δ+1 , where δ 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 diam(G) ≥ 3 then we further improve the bound to πopt(G) ≤ 3.75n δ+1. On the other hand, we show that a family of graphs with optimal pebbling number 8n 3(δ+1) exists.

Discrete Mathematics

Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs.

International Journal of Game Theory, 2021

A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root . The pebbling number of a graph $$\pi (G)$$ π ( G ) is the minimum number of pebbles needed so every distribution of $$\pi (G)$$ π ( G ) pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations of various classes of graphs for which each player has a winning strategy. We will characterize the winning player for a specific class of diameter two graphs.

Discrete Applied Mathematics, 2013

Graph pebbling is the study of whether pebbles from one set of vertices can be moved to another while pebbles are lost in the process. A number of variations on the theme have been presented over the years. In this paper we provide a common framework for studying them all, and present the main techniques and results. Some new variations are introduced as well and open problems are highlighted.

2004

We say that a graph G is Class 0 if its pebbling number is exactly equal to its number of vertices. For a positive integer d, let k(d) denote the least positive integer so that every graph G with diameter at most d and connectivity at least k(d) is Class 0. The existence of the function k was conjectured by Clarke, Hochberg and Hurlbert, who showed that if the function k exists, then it must satisfy k(d)=\Omega(2^d/d). In this note, we show that k exists and satisfies k(d)=O(2^{2d}). We also apply this result to improve the upper bound on the random graph threshold of the Class 0 property.

arXiv: Combinatorics, 2019

Graph pebbling models the transportation of consumable resources. As two pebbles move across an edge, one reaches its destination while the other is consumed. The $t$-pebbling number is the smallest integer $m$ so that any initially distributed supply of $m$ pebbles can place $t$ pebbles on any target vertex via pebbling moves. The 1-pebbling number of diameter two graphs is well-studied. Here we investigate the $t$-pebbling number of diameter two graphs under the lense of connectivity.