The cover pebbling number of the join of some graphs

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

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.

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.

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.

Arxiv preprint math/ …, 2005

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. We introduce the notion of domination cover pebbling, obtained by ...

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 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.

Asian-European Journal of Mathematics

A new graph invariant called the secure vertex cover pebbling number, which is a combination of two graph invariants, namely, ‘secure vertex cover’ and ‘cover pebbling number’, is introduced in this paper. The secure vertex cover pebbling number of a graph, [Formula: see text], is the minimum number [Formula: see text] so that every distribution of [Formula: see text] pebbles can reach some secure vertex cover of [Formula: see text] by a sequence of pebbling moves. In this paper, the complexity of the secure vertex cover problem and secure vertex cover pebbling problem are discussed. Also, we obtain some basic results and the secure vertex cover pebbling number for complete [Formula: see text]-partite graphs, paths, Friendship graphs, and wheel graphs.

Discrete Applied Mathematics

A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number π opt is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. A rubbling move is similar to a pebbling move, but it can remove the two pebbles from two different vertex. The optimal rubbling number ρ opt is defined analogously to the optimal pebbling number. In this paper we give lower bounds on both the optimal pebbling and rubbling numbers by the distance k domination number. With this bound we prove that for each k there is a graph G with diameter k such that ρ opt (G) = π opt (G) = 2 k .