Pebbling in Split Graphs

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.

💡 Research Summary

Graph pebbling is a discrete‐resource transportation model in which moving two pebbles across an edge consumes one pebble. The pebbling number π(G) of a graph G is the smallest integer t such that, from any placement of t pebbles on the vertices, a pebble can be moved to any prescribed target vertex using pebbling steps. Determining whether a given configuration can reach a target is NP‑complete, and deciding whether π(G) ≤ k is Π₂^P‑complete. Nevertheless, exact formulas or polynomial‑time algorithms are known for many families (complete graphs, trees, cycles, hypercubes, diameter‑2 graphs, etc.).

This paper focuses on split graphs, a subclass of chordal graphs of diameter three. A split graph’s vertex set can be partitioned into a clique K and an independent set S (the “cones”). The authors derive a closed‑form expression for the pebbling number of any split graph and present an algorithm that computes it in O(n^β) time, where β = 2ω/(ω+1) ≈ 1.41 and ω ≈ 2.376 is the matrix‑multiplication exponent. They also prove that every split graph with minimum degree at least three is Class 0 (π(G)=|V(G)|).

The technical core is the Weight‑Function Lemma (Lemma 1), originally introduced in earlier work. For a rooted subtree T of G, a weight function w assigns non‑negative integers to vertices such that w(root)=0 and w(parent) ≥ 2·w(child) for every non‑root edge. If a configuration C is r‑unsolvable, then w(C) ≤ w(T). By constructing several such strategies (subtrees) and summing their weight functions, one obtains a global weight function w satisfying w(v) ≥ 1 for all v ≠ r. Consequently, any unsolvable configuration must have size at most w(T), yielding the upper bound π(G,r) ≤ ⌊w(T)⌋+1.

Using this lemma, the authors treat three cases:

1. Target r in the clique K.
   Let X be the set of cut vertices of G and x = |X|. They show π(G,r) = n + x, where n = |V(G)|. The lower bound comes from a configuration that places three pebbles on each cut vertex and one pebble elsewhere; the upper bound follows from a collection of strategies that give weight 2 to each cut vertex and weight 1 to all other vertices. Moreover, every pebbling step in an optimal solution is greedy (distance to r never increases).

2. Target r in the independent set S with eccentricity 2.
   Here the graph may contain a special substructure called an r‑Pereyra (three cone vertices of degree 2 whose neighborhoods do not satisfy the Helly property). If G is r‑Pereyra, then π(G,r) = n + 1; otherwise π(G,r) = n. The extra “+1” term is denoted ψ(G,r). The proof uses a detailed analysis of unsolvable configurations and shows that any solution can be made r‑semigreedy (each move does not increase the distance to r).

3. Target r in S with eccentricity 3.
   This is the most intricate case. Vertices are classified as “good cones” or “bad cones” depending on whether removing them leaves an r‑Pereyra graph. Multiple strategies are built: each cone contributes weight 2 in two strategies, each non‑cone vertex contributes weight 1, and vertices adjacent to cones receive fractional weight ½ in two strategies. Summing all strategies yields weight 2 on every cut vertex and weight 1 elsewhere, leading again to π(G,r) = n + x + ψ, where ψ = 1 precisely when G is an r‑Pereyra or an r‑Phoenix (eccentricity 3 and a minimum degree condition δ* ≥ 4). The authors prove that any unsolvable configuration must violate the weight bound, establishing the exact formula.

The algorithmic implementation follows directly from the construction of the strategies. Computing the necessary neighborhoods and identifying cut vertices can be done in linear time; the aggregation of weight functions reduces to matrix‑multiplication‑type operations, giving the overall running time O(n^{2ω/(ω+1)}) ≈ O(n^{1.41}). This improves dramatically over naïve O(n²) or O(n³) approaches.

Finally, the paper shows that if a split graph has minimum degree δ(G) ≥ 3, then it contains no cut vertices (x = 0), so π(G) = n for every target. Hence all such graphs belong to Class 0. The authors also provide linear‑time recognition procedures for the exceptional Pereyra and Phoenix structures, completing the characterization.

In summary, the paper delivers:

• A precise pebbling‑number formula for every split graph, distinguishing three target‑type cases and an extra term ψ for the rare Pereyra/Phoenix configurations.
• A novel application of the weight‑function technique to obtain tight upper bounds.
• An O(n^{1.41}) algorithm for computing π(G) on split graphs, the first sub‑quadratic method for this class.
• A structural result that all split graphs with minimum degree at least three are Class 0.

These contributions advance both the theoretical understanding of pebbling numbers in chordal graphs and the practical ability to compute them efficiently.