Pebbling in Split Graphs

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 minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter three chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(n^\beta) time, where \beta=2\omega/(\omega+1)\cong 1.41 and \omega\cong 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.