Pebbling in Semi-2-Trees
  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. 
Liliana Alc\'on, Marisa Gutierrez, and Glenn Hurlbert
math.CO