Pebbling Numbers in Graph Theory

• Graph pebbling is a discrete optimization framework where pebbles are moved across vertices with consumption rules to determine the t-pebbling number needed for guaranteed resource delivery.
• It defines moves that remove two pebbles from one vertex to send one to an adjacent vertex, capturing worst-case resource scenarios through extremal and superstack configurations.
• Algorithmic techniques like maximum path partitions and Chung configurations enable polynomial-time computation on trees, offering insights into broader conjectures in graph theory.

Graph pebbling is a discrete optimization paradigm centered around transporting finite, consumable resources ("pebbles") across the vertices of a graph according to prescribed local loss rules. The t-pebbling number is a core parameter, measuring the minimal resource supply needed to guarantee transfer of $t$ pebbles to a specified vertex from any initial configuration, via pebbling moves that remove two pebbles from a vertex to send one to an adjacent vertex (consuming one in transit). The concept generalizes to arbitrary demand distributions over vertices and plays a central role in understanding worst-case resource requirements, the structure of extremal distributions, and algorithmic methods in graph theory.

1. Formal Definition and Generalizations

Let $G = (V, E)$ be a connected undirected graph. A configuration $C: V \rightarrow \mathbb{N}$ represents an allocation of pebbles. A pebbling move from $u$ to $v$ (with $uv \in E$) subtracts two pebbles from $u$ ($C(u) \geq 2$ required) and adds one pebble to $v$. For a demand (distribution) $D: V \rightarrow \mathbb{N}$, a configuration $C$ solves $D$ if some sequence of pebbling moves yields $C'$ with $C'(v) \geq D(v)$ for all $v$.

• The $D$-pebbling number $\pi(G,D)$ is the smallest integer $m$ such that every configuration $C$ with $|C| \geq m$ solves $D$.
• When $D = r^t$ (i.e., $D(r) = t$, all other entries $0$), the $t$-fold pebbling number is $\pi_t(G) = \max_{r \in V} \pi(G, r^t)$.

Key generalizations include cover pebbling (demands positive on all $v$), fractional pebbling (continuous pebbles and moves), and extension to arbitrary demand distributions. The support of a configuration or distribution is $\{ v \mid F(v) > 0 \}$.

2. Extremal Configurations in Trees

For trees, the structure of maximal unsolvable configurations is precisely characterized. Given a tree $T$, any configuration of maximal size not solving $D$ (called a $D$-extremal configuration) must place nonzero pebbles only on the leaves of $T$. No such configuration places pebbles in the interior; as a result, extremal configurations are "stacked" on leaves.

A "superstack" (Editor's term) is a leaf with the property that adding one pebble renders the configuration $D$-solvable. In fact, every $D$-extremal configuration in a tree can be realized as a superstack at some leaf. This generalizes earlier results for single-target demands and provides a canonical witness for extremal unsolvability.

3. Polynomial-Time Computation in Trees

The main result is a polynomial-time algorithm for computing $\pi(T, D)$ for any tree $T$ and demand $D$:

• Let $T(D)$ be the minimal (convex hull) subtree containing all vertices demanded by $D$.
• Let $L$ be the set of leaves of $T(D)$.
• For each $v_i \in L$, compute its local demand $$d'_i = \sum_{u \in \mathrm{supp}(D)} D(u) \cdot 2^{d(v_i,u)}$$ ($d(v_i,u)$ is the distance in $T$).
• For each $v_i$, construct corresponding auxiliary subtrees and compute their maximum path partition (ordering as in majorization).
• The D-extremal configuration for $v_i$ is the "Chung configuration": assign $d'_i - 1$ pebbles to $v_i$ and suitable numbers to leaves of the partition as in Chung's formula.
• Finally,

$$\pi(T, D) = \max_{v \in L} \{ \text{size of Chung configuration at } v \} + 1.$$

The total running time is $O(s(D) n)$ for $n$-vertex $T$ and $s(D) = |\mathrm{supp}(D)|$.

This algorithm also yields $\pi_t(T)$ in $O(n^2)$ time for t-fold pebbling. Previous results (e.g. Chung's 1989 theorem) are recovered as special cases.

4. Connection to Path Partitions and Chung's Theorem

The algorithm is rooted in path partition theory. A maximum path partition is a set of vertex-disjoint directed paths in $T$ whose sequence of lengths majorizes any other such partition. For a single-target $r^t$, the Chung configuration assigns $t \cdot 2^{a_1} - 1$ pebbles at the designated farthest leaf (with $a_1$ the longest path), $2^{a_i} - 1$ at leaves of the other (shorter) paths, and zero elsewhere. This method extends to general $D$ by superposition of Chung configurations at multiple targets and their convex hull.

The algorithm's correctness and structure theory confirm the optimality of this construction, both for single and multiple targets, establishing a tight link between extremal configurations and structural decompositions of trees.

5. Broader Theoretical and Algorithmic Implications

The characterization and algorithm for trees form a critical step in attack on the more general Pyramid-Free Chordal Graph Pebbling Number Conjecture (Alcon, Gutierrez, Hurlbert). This conjecture posits that for every chordal graph without an induced pyramid, the $D$-pebbling number can be computed in polynomial time, and that extremal configurations are supported on the set of simplicial vertices (which are leaves in trees). The structure results in trees provide strong evidence supporting this conjecture and suggest potential extension to classes such as split graphs and interval graphs.

Furthermore, the computation of $\pi(T, D)$ rigorously supports the Weak and Strong Target Conjectures, which posit precise relationships between the pebbling number for arbitrary demands and the t-fold pebbling number, in specific graph families including trees.

6. Relationships to Other Pebbling Parameters

The results relate directly to cover pebbling (Xia et al., 2019), t-pebbling (0905.3949), and general $D$-pebbling analysis (Alcón et al., 2021). The formula for $\pi(T, D)$ can be interpreted as a maximal value over leaf-based "superstack" configurations, reinforcing the paradigm that, in trees, extremal behaviors are governed by supply concentrated on peripheral vertices.

The approach also fits within the wider context of weight function and linear optimization techniques (Li, 14 Sep 2025, Hurlbert, 2011), where pebbling numbers are characterized by maximizing total weight under pebbling-imposed constraints. In trees, the explicit "stacked on leaves" witness configurations eliminate the need for further linear optimization or auxiliary polytopal machinery.

7. Applications and Open Problems

This explicit, efficiently constructible characterization supports applications in network reliability, worst-case resource allocation, and analysis of dynamic processes on tree networks.

The most immediate open problems include:

• Extension of the leaf-concentration property to wider chordal or pyramid-free graph classes.
• Proving or disproving the No-Merging Lemma for chordal graphs and understanding the potential generality of superstack structure for extremal configurations.
• Developing polynomial-time algorithms for $D$-pebbling numbers in broader graph families, guided by tree analogues.

This framework significantly deepens the combinatorial theory and algorithmic tractability of graph pebbling in trees, provides structural models likely to extend to larger graph classes, and illuminates foundational connections among key pebbling invariants, resource transportation, and extremal configuration analysis (Adauto et al., 14 Apr 2025).