Related papers: Stacking and clearing in graph pebbling
Here we introduce a new game on graphs, called cup stacking, following a line of what can be considered as $0$-, $1$-, or $2$-person games such as chip firing, percolation, graph burning, zero forcing, cops and robbers, graph pebbling, and…
A pebbling move on a graph consists of taking two pebbles off from one vertex and add one pebble on an adjacent vertex, the $t$-pebbling number of a graph $G$ is the minimum number of pebbles so that we can move $t$ pebbles on any vertex on…
Consider a distribution of pebbles on a graph. A pebbling move removes two pebbles from a vertex and place one at an adjacent vertex. A vertex is reachable under a pebble distribution if it has a pebble after the application of a sequence…
Recent research in graph pebbling has introduced the notion of a cover pebbling number. Along this same idea, we develop a more general pebbling function Pi(G, t, P). This measures the minimum number of pebbles needed to guarantee that any…
Given a graph G and a configuration C of pebbles on the vertices of G, a pebbling step removes two pebbles from one vertex and places one pebble on an adjacent vertex. The cover pebbling number g=g(G) is the minimum number so that every…
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 $\pi(G)$ is the smallest number such that, for any…
The $t$-fold pebbling number, $\pi_t(G)$, of a graph $G$ is defined to be the minimum number $m$ so that, from any given configuration of $m$ pebbles on the vertices of $G$, it is possible to place at least $t$ pebbles on any specified…
Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…
Given a configuration of pebbles on the vertices of a graph $G$, a pebbling move removes two pebbles from a vertex and puts one pebble on an adjacent vertex. The pebbling number of a graph $G$ is the smallest number of pebbles required such…
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…
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…
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. The pebbling number $\pi(G)$ is the smallest $t$ so that from any…
Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at…
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…
In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like ``pegging moves'' allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging…
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…
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 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…
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…
In the game of pegging, each vertex of a graph is considered a hole into which a peg can be placed. A pegging move is performed by jumping one peg over another peg, and then removing the peg that has been jumped over from the graph. We…