Restricted optimal pebbling and domination in 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.

2011

We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of pebbles on the vertices. Player one wins if, through a series of pebbling moves, he can move at

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 ...

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.

2015

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 optimal pebbling (rubbling) number is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable using pebbling (rubbling) moves. We determine the optimal rubbling number of ladders ($P_n\square P_2$), prisms ($C_n\square P_2$) and M\"oblus-ladders. We also give upper and lower bounds for the optimal pebbling and rubbling numbers of large grids ($P_n\square P_n$).

Graphs and Combinatorics

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 $$\pi _{{{\,\mathrm{opt}\,}}}$$πopt is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph $$P_n\square P_m$$Pn□Pm was investigated in several papers (Bunde et al. in J Graph Theory 57(3):215–238, 2008; Xue and Yerger in Graphs Combin 32(3):1229–1247, 2016; Győri et al. in Period Polytech Electr Eng Comput Sci 61(2):217–223 2017). In this paper, we present a new method using some recent ideas to give a lower bound on $$\pi _{{{\,\mathrm{opt}\,}}}$$πopt. We apply this technique to prove that $$\pi _{{{\,\mathrm{opt}\,}}}(P_n\square P_m)\ge \frac{2}{13}nm$$πopt(Pn□Pm)≥213nm. Our method also gives a new proof for $$\pi _{{{\,...

Journal of Combinatorial Optimization, 2016

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move removes two pebbles from some vertex and places one pebble 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. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every n-vertex Class 0 graph has at least 5 3 n− 11 3 edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to 2n − 5, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.

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.

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...

2019

Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number π(G) is the smallest number such that, for any distribution of π(G) pebbles to the vertices of G and choice of root vertex r of G, there exists a sequence of pebbling moves that places a pebble on r. Computing π(G) is provably difficult, and recent methods for bounding π(G) have proved computationally intractable, even for moderately sized graphs. Graham conjectured that π(G H) ≤ π(G)π(H), where G H is the Cartesian product of G and H (1989). While the conjecture has been verified for specific families of graphs, in general it remains open. This study combines the focus of developing a computationally tractable, IP-based method for generating good bounds on π(G H), with the goal of shedding light on Graham’s conjecture.We provide computational results for a variety of Cartesian-product graphs, inclu...

arXiv: Combinatorics, 2021

Graph pebbling is a network optimization model for satisfying vertex demands with vertex supplies (called pebbles), with partial loss of pebbles in transit. The pebbling number of a demand in a graph is the smallest number for which every placement of that many supply pebbles satisfies the demand. The Target Conjecture (Herscovici-Hester-Hurlbert, 2009) posits that the largest pebbling number of a demand of fixed size $t$ occurs when the demand is entirely stacked on one vertex. This truth of this conjecture could be useful for attacking many open problems in graph pebbling, including the famous conjecture of Graham (1989) involving graph products. It has been verified for complete graphs, cycles, cubes, and trees. In this paper we prove the conjecture for 2-paths and Kneser graphs over pairs.

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.

Eprint Arxiv 0907 5577, 2009

We prove a generalization of Graham's Conjecture for optimal pebbling with arbitrary sets of target distributions. We provide bounds on optimal pebbling numbers of products of complete graphs and explicitly find optimal $t$-pebbling numbers for specific such products. We obtain bounds on optimal pebbling numbers of powers of the cycle $C_5$. Finally, we present explicit distributions which provide asymptotic bounds on optimal pebbling numbers of hypercubes.