Optimal pebbling and rubbling of graphs with given diameter

Discrete Applied Mathematics, 2017

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.

Electronic Notes in Discrete Mathematics, 2011

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.

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...

Journal of Graph Theory, 2008

Given a distribution of pebbles on the vertices of a graph G, a pebbling move takes two pebbles from one vertex and puts one on a neighboring vertex. The pebbling number Π(G) is the least k such that for every distribution of k pebbles and every vertex r, a pebble can be moved to r. The optimal pebbling number Π OP T (G) is the least k such that some distribution of k pebbles permits reaching each vertex.

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.

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.

Integers, 2000

Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one of them to a neighboring vertex. The player wins the game if he can move k pebbles to r. The value of the game (G, k), called the k-pebbling number of G and denoted π k (G), is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, from every configuration of size m, one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the k-pebbling number of diameter-two graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two.

Electronic Notes in Discrete Mathematics, 2009

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. We improve on the bound of Bukh by showing that the pebbling number of a graph of diameter three on n vertices is at most 3n/2 + 2, and this bound is best possible. We obtain an asymptotically best possible bound of 3n/2 + Θ(1) for the pebbling number of graphs of diameter four. Finally, we prove an asymptotic upper bound for the pebbling number of graphs of diameter d, namely (2 d 2 − 1)n + O(1), and this also improves a bound given by Bukh.

2004

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, gamma(G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on