Pebbling in Split Graphs
Liliana Alc\'on, Marisa Gutierrez, and Glenn Hurlbert

TL;DR
This paper derives a formula and efficient algorithm for computing the pebbling number of split graphs, a family of diameter three chordal graphs, and shows that all split graphs with minimum degree at least 3 are Class 0.
Contribution
It provides the first explicit formula and polynomial-time algorithm for pebbling numbers in split graphs, and characterizes when they are Class 0 based on minimum degree.
Findings
Formula for pebbling number of split graphs
Algorithm with O(n^β) complexity for pebbling number calculation
All split graphs with minimum degree ≥ 3 are Class 0
Abstract
Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of two pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding if a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and that deciding if the pebbling number has a prescribed upper bound is \Pi_2^P-complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having…
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Taxonomy
TopicsInterconnection Networks and Systems · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
