Pebbling in Semi-2-Trees

TL;DR

This paper investigates the pebbling number in semi-2-trees, a subclass of chordal graphs, providing a polynomial-time formula that advances understanding of pebbling complexity in graphs with small tree width and large diameter.

Contribution

It introduces a polynomial-time computable formula for the pebbling number of semi-2-trees, extending previous results beyond split graphs to a broader class of chordal graphs.

Findings

Pebbling number of semi-2-trees can be computed in polynomial time.

Provides a formula for pebbling number specific to semi-2-trees.

Advances the classification of graph classes with efficiently computable pebbling numbers.

Abstract

Graph pebbling is a network model for transporting discrete resources that are consumed in transit. Deciding whether a given configuration on a particular graph can reach a specified target is ${\sf NP}$-complete, even for diameter two graphs, and deciding whether the pebbling number has a prescribed upper bound is $\Pi_2^{\sf P}$-complete. Recently we proved that the pebbling number of a split graph can be computed in polynomial time. This paper advances the program of finding other polynomial classes, moving away from the large tree width, small diameter case (such as split graphs) to small tree width, large diameter, continuing an investigation on the important subfamily of chordal graphs called $k$-trees. In particular, we provide a formula, that can be calculated in polynomial time, for the pebbling number of any semi-2-tree, falling shy of the result for the full class of 2-trees.