2-Connected

the property that a graph cannot be disconnected by removing only one vertex

2-Pebbling Property

two pebbles can be moved to any specified vertex when the initial configuration $C$ has size $2\pi(G)-s(C)+1$, where $s(C)$ is the size of the support

Bipartite Graph

a graph whose vertices can be covered by two sets of pairwise nonadjacent vertices

Book

the graph $S_{p+1} \times P_q$, having $p$ pages, $q$ vertices per page, and $q$ vertices on the binding.

Cartesian Product

G₁ x G₂ has vertex set $V(G_1 \times G_2) = \{(v_1,v_2) | v_1 \in V(G_1), v_2 \in V(G_2) \}$ and edge set $E(G_1 \times G_2) = \{\{(v_1,v_2),(w_1,w_2)\} | (v_1=w_1$ $and$ $(v_2,w_2) \in E(G_2))$ $or$ $(v_2=w_2$ $and$ $(v_1,w_1) \in E(G_1))\}$

Class 0

the property that a graph has pebbling number equal to its number of vertices

Complete

the property that a graph has every vertex adjacent to all other vertices

Complete Bipartite Graph

a bipartite graph in which every pair of vertices not belonging to the same partite set is adjacent

Configuration

a function that indicates the number of pebbles per vertex

Connectivity

the minimum number of vertices whose deletion disconnects the graph or reduces it to one vertex

Cover Pebbling Number

the minimum number of pebbles needed so that from any initial configuration of the pebbles, after a series of pebbling moves, it is possible to have at least $1$ pebble on every vertex of a graph

Cover Pebbling Ratio

for a graph $G$, the cover pebbling number of $G$ divided by the pebbling number of $G$

Cross Number

for a sequence $S$ in a group, the value: $\displaystyle{\sum_{g \in S}\frac{1}{|g|}}$

Cube

a graph that has all binary $d$-tuples as vertices and edges between that differ in exactly one coordinate

Cycle

a simple graph whose vertices can be placed on a circle so that vertices are adjacent if and only if they appear consecutively on the circle

Davenport's Constant

the smallest $D$ such that every sequence of $D$ elements contains a zero-sum subsequence

Diameter

the maximum distance between two vertices of a graph

Dominating Set

a subset $S$ of the vertex set of a graph such that every vertex outside of $S$ has a neighbor in $S$

Domination Number

the minimum size of a dominating set of vertices

Fractional Pebbling Number

$\liminf_{k \rightarrow \infty} \pi(G)/k$

Fuse

the graph on $n$ vertices composed of a path on $l$ vertices with $n-l$ independent vertices incident to one of its endpoints

Girth

the length of a shortest cycle in a graph

Greedy Graph

a graph $G$ for which every configuration of size $\pi(G)$ has a greedy slolution

Greedy Solution

a pebbling solution in which every step is greedy

Greedy Step

a pebbling step from a vertex $u$ to a vertex $v$ such that ${\rm dist}(v,r)<{\rm dist}(u,r)$, where $r$ is the root vertex

H-Sum Sequence

for a subgroup $H$ of a group $G$, a sequence of elements of $G$ that sums to an element of $H$

Kneser Graph

the graph with vertices ${[m] \choose t}$ and edges $\{A,B\}$ whenever $A$ and $B$ are disjoint

Majorize

the sequence $(s_1,\ldots,s_k)$ majorizes the sequence $(t_1,\ldots,t_k)$ if $s_i > t_i$ when $s_j=t_j$ for all $j

Maximum Path Partition

a path partition of a tree $T$ such that no other path partition majorizes it

Optimal Pebbling Number

the size of the smallest solvable configuration in a graph

Path

a sequence of vertices, each adjacent to its successor

Path Partition

an $r$-path partition for some $r$

Pebbling Number

the fewest number of pebbles that gurantee that a specific graph is solvable

Pebbling Threshold

a threshold for the solvability of random pebbling configurations on graph sequences

Petersen Graph

the disjointness graph of the $2$-sets in a $5$-element set

Positive Weight Function

$w(v)>0$ for every vertex $v$, where $w(v)$ is the number of pebbles on vertex $v$

$q$-Pebbling Step

consists of removing $q$ pebbles from a vertex $u$, and placing one pebble on a neighbor $v$ of $u$

$r$-Maximum Path Partition

an $r$-path partition of a tree $T$ such that no other $r$-path partition majorizes it

$r$-Path Partition

a partition $Q=(Q_1,\ldots,Q_m)$ of the edges of a tree $T$ into paths such that each path $Q_i$ is well $r$-directed

r-Solvable

the property that there is a sequence of pebbling moves for a given graph such that it is possible to place a pebble on the root

r-Unsolvable

the property that for a given graph, it is not possible to place a pebble on the root through a sequence of pebbling moves

Root

the pebbling target

Semi-greedy

a pebbling step from a vertex $u$ to a vertex $v$ such that ${\rm dist}(v,r) \leq {\rm dist}(u,r)$, where $r$ is the root vertex

Simple Configuration

all pebbles are concentrated on a single vertex

Solvable

a configuration that is $r$-solvable for every root $r$

Star

a graph made up of one vertex adjacent to each of $n$ independent vertices

Support

the set of all vertices that have at least one pebble

Threshold

for a property $\mathbf{P}$ of a random structure $S_p$, the set of functions $t$ for which $p \gg t$ implies that $Pr[S_p$ $has$ $\mathbf{P}] \rightarrow 0$ as $n \rightarrow \infty$, and $p \ll t$ implies that $Pr[S_p$ $has$ $\mathbf{P}] \rightarrow 1$ as $n \rightarrow \infty$

Tree

a connected graph with no cycles

Tree-Solvable

the property that every configuration of size at least $\pi(G)$ has a solution in which the edges traversed by pebbling steps form an acyclic subgraph

Weighted Cover Pebbling Number

the minimum number of pebbles needed to place, after a sequence of pebbling steps, $w(v)$ pebbles on vertex $v$, for all $v$, regardless of the initial configuration

Weighted Pebbling Number

the minimum number $t$ so that, for every weight function $w$ with $|w|=\mathbf{w}$, every configuration of $t$ pebbles is $w$-solvable

Well $r$-Directed

An orientatiuon of the edges of a path $P$ with one endpoint $r$ for which every edge points to $r$

Zero-Sum Sequence

a sequence of elements of a finite group $G$ that sums to the identity of $G$