Related papers: Approximating Cumulative Pebbling Cost is Unique G…
We consider the computational complexity of finding a legal black pebbling of a DAG $G=(V,E)$ with minimum cumulative cost. A black pebbling is a sequence $P_0,\ldots, P_t \subseteq V$ of sets of nodes which must satisfy the following…
Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\vert f\vert=\sum_{u\in V}f(u)$ is called the weight of $f$ which is just the…
Pebble games were originally formulated to study time-space tradeoffs in computation, modeled by games played on directed acyclic graphs (DAGs). Close connections between pebbling and cryptography have been known for decades. A series of…
It is shown that $S(G) = O\left(m/\log_2 m + d\right)$ pebbles are sufficient to pebble any DAG $G=(V,E)$, with $m$ edges and maximum in-degree $d$. It was previously known that $S(G) = O\left(d n/\log n\right)$. The result builds on two…
Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the weight of $f$ is $w(f)=\sum_{u\in V}f(u)$ which is just the total number of pebbles…
Let $G=(V,E)$ be a simple graph. A function $\phi:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\sum_{u\in V}\phi(u)$ is called the size of $\phi$ which is just the total…
We study the approximability of a number of graph problems: treewidth and pathwidth of graphs, one-shot black (and black-white) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as minimum cut…
We explore the complexity of computing the optimal pebbling number and pebbling number of a graph. We show that deciding whether the optimal pebbling number of G is at most k is NP-complete and deciding whether the pebbling number of G is…
The pebbling number of a graph $G$, $f(G)$, is the least $p$ such that, however $p$ pebbles are placed on the vertices of $G$, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and…
Let $G=(V,E)$ be a simple graph. A pebbling configuration on $G$ is a function $f:V\rightarrow \mathbb{N}\cup \{0\}$ that assigns a non-negative integer number of pebbles to each vertex. The weight of a configuration $f$ is $w(f)=\sum_{u\in…
Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble…
Pebbling on graphs is a two-player game which involves repeatedly moving a pebble from one vertex to another by removing another pebble from the first vertex. The pebbling number $\pi(G)$ is the least number of pebbles required so that,…
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…
Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. A configuration $C$ is a supply of pebbles at various vertices of a…
Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The pebbling…
A pebbling move on a weighted graph removes some pebbles at a vertex and adds one pebble at an adjacent vertex. The number of pebbles removed is the weight of the edge connecting the vertices. A vertex is reachable from a pebble…
Pebble games are single-player games on DAGs involving placing and moving pebbles on nodes of the graph according to a certain set of rules. The goal is to pebble a set of target nodes using a minimum number of pebbles. In this paper, we…
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…
This paper discusses the complexity of graph pebbling, dealing with both traditional pebbling and the recently introduced game of cover pebbling. Determining whether a configuration is solvable according to either the traditional definition…
Consider a configuration of pebbles distributed on the vertices of a connected graph of order $n$. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles…