Target Pebbling in Trees

TL;DR

The paper addresses computing target pebbling numbers $π(G,D)$, with a focus on trees. It develops a polynomial-time algorithm for $π(T,D)$ on any tree $T$ by leveraging path partitions, Chung-type configurations, and structural lemmas such as the No-Cycle and No-Merging properties, culminating in a leaf-supported extremal configuration characterization via superstacks. A key contribution is a closed-form framework for trees (and the associated algorithm running in $O(s(D)·n)$ time) that yields $π(T,D)$ and extends known results for $π_t(T)$. These results advance exact pebbling computations for trees, with potential implications for pyramid-free chordal graphs and broader pebbling conjectures, including Graham-type product bounds. The work tightens the link between combinatorial structure and pebbling feasibility, enabling efficient and precise analysis of pebbling demands on trees.

Abstract

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 $G$, and a distribution $D$ is a demand of pebbles at various vertices of $G$. The $D$-pebbling number, $π(G, D)$, of a graph $G$ is defined to be the minimum number $m$ such that every configuration of $m$ pebbles can satisfy the demand $D$ via pebbling moves. The special case in which $t$ pebbles are demanded on vertex $v$ is denoted $D=v^t$, and the $t$-fold pebbling number, $π_{t}(G)$, equals $\max_{v\in G}π(G,v^t)$. It was conjectured by Alcón, Gutierrez, and Hurlbert that the pebbling numbers of chordal graphs forbidding the pyramid graph can be calculated in polynomial time. Trees, of course, are the most prominent of such graphs. In 1989, Chung determined $π_t(T)$ for all trees $T$. In this paper, we provide a polynomial-time algorithm to compute the pebbling numbers $π(T,D)$ for all distributions $D$ on any tree $T$, and characterize maximum-size configurations that do not satisfy $D$.

Figures (4)

• Figure 1: A path $P$ with both endpoints in $D$, showing the sequence of configurations $C$, $C'$, ..., and $C_n$, (in green), and targets $D$, $D'$, ..., and $D_R^+$ (in red) that are used in the proof of Theorem \ref{['t:Path']}.
• Figure 2: A tree $T$ with target $D$ (in red) and $D$-extremal configurations (in green) $C$ (on the left) and $C'$ (on the right), illustrating Lemma \ref{['l:NoMerge']}. Both $w$ and $j$ are merging vertices of the $(C+y,D)$-solution ${\sigma}$, with $w$ on the $jy$-path of $T_{\sigma}$. Then the argument in the proof of Lemma \ref{['l:NoMerge']} converts $C$ to $C'$ by moving $2(2^2)=8$ pebbles from $x$ to $y$.
• Figure 3: A tree $T$ with bushes $B_u$ (in blue), $B_x$ (in green), $B_w$ (in brown), and $B_v$ (in orange), above, and the tree $T$, redrawn with choice of $\prec$-minimal bush $B_v$, below.
• Figure 4: The tree $T'=B_u\cup B_v\cup T_{uv}$ (derived from Figure \ref{['fig:bush1']}), showing bushes $B_{z_u}$ (in blue) and $B_{z_v}$ (in orange), with corresponding seeds $z_u$ and $z_v$, above, and the pebbling function $D'$ on $T_{uv}$.

Theorems & Definitions (28)

• Conjecture 2: AlcoHurlPowers
• Theorem 3: Chung Chung
• Lemma 4: No-Cycle Lemma Moews
• Theorem 6: Alcón, Hurlbert AlcoHurlPowers
• Conjecture 7
• Theorem 8: S05VW04
• Conjecture 9: Weak Target Conjecture
• Conjecture 10: Strong Target Conjecture
• Theorem 11
• Theorem 12