Related papers: Target Pebbling in Trees
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…
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 this paper, we study the treasure hunt problem in a graph by a mobile agent. The nodes in the graph $G=(V,E)$ are anonymous and the edges incident to a vertex $v\in V$ whose degree is $deg(v)$ are labeled arbitrarily as $0,1,\ldots,…
Let $G$ be a connected graph. A pebbling move is defined as taking two pebbles from one vertex and placing one pebble to an adjacent vertex and throwing away the other pebble. The non-split domination cover pebbling number, $\psi_{ns}(G)$,…
Graph Pebbling is a well-studied single-player game on graphs. We introduce the game of Blocking Pebbles which adapts Graph Pebbling into a two-player strategy game in order to examine it within the context of Combinatorial Game Theory.…
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. A function is a pebbling threshold for a sequence of graphs if a randomly chosen…
Given a connected, undirected, simple graph $G = (V, E)$ and $p \le |V|$ pebbles labeled $1,..., p$, a configuration of these $p$ pebbles is an injective map assigning the pebbles to vertices of $G$. Let $S$ and $D$ be two such…
Given a connected graph $G$ and a configuration of $t$ pebbles on the vertices of G, a $q$-pebbling step consists of removing $q$ pebbles from a vertex, and adding a single pebble to one of its neighbors. Given a vector…
The restricted edge pebbling distribution is a distribution of pebbles on the edges of $G$ is the placement of pebbles on the edges with the restriction that only an even number of pebbles should be placed on the edges with labels $0$.…
We survey results on the pebbling numbers of graphs as well as their historical connection with a number-theoretic question of Erd\H os and Lemke. We also present new results on two probabilistic pebbling considerations, first the random…
In a graph $G$, we define a set of vertices to be a \emph{strong hub set} if for any two vertices in $G$, we can find a path between them whose internal vertices are all in this set. We define the \emph{strong hub cover pebbling number} of…
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…
We define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of the…
Cubicity of a graph $G$ is the smallest dimension $d$, for which $G$ is a unit disc graph in ${\mathbb{R}}^d$, under the $l^\infty$ metric, i.e. $G$ can be represented as an intersection graph of $d$-dimensional (axis-parallel) unit…
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 is the smallest $t$ so that from any initial…
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…
We study the problem of planning paths for $p$ distinguishable pebbles (robots) residing on the vertices of an $n$-vertex connected graph with $p \le n$. A pebble may move from a vertex to an adjacent one in a time step provided that it…
In this note we answer a question of Hurlbert about pebbling in graphs of high girth. Specifically we show that for every g there is a Class 0 graph of girth at least g. The proof uses the so-called Erdos construction and employs a recent…
Optimizing data movements during program executions is essential for achieving high performance in modern computing systems. This has been classically modeled with the Red-Blue Pebble Game and its variants. In existing models, it is…
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…