Pebbling in Split Graphs

Abstract

Abstract. Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of 2 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 whether a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter 2 graphs, and that deciding whether the pebbling number has a prescribed upper bound is Π P 2 -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 2 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 3 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 β ) time, where β = 2ω/(ω + 1) ∼ = 1.41 and ω ∼ = 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.Key words. pebbling number, split graphs, class 0, graph algorithms, complexity AMS subject classifications. 05C85, 68Q17, 90C35 DOI. 10.1137/130914607 1. Introduction. Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: while 2 pebbles cross an edge of a graph, only one arrives at the other end as the other is consumed (or spent on toll, one can imagine). This operation is called a pebbling step. The basic questions in the subject revolve around deciding whether a particular configuration of pebbles on the vertices of a graph can reach a given root vertex via pebbling steps (for this reason, graph pebbling is carried out on connected graphs only). If a configuration can reach r, it is called r-solvable, and r-unsolvable otherwise.Various rules for pebbling steps have been studied for years and have found applications in a wide array of areas. One version, dubbed black and white pebbling, was applied to computational complexity theory in studying time-space tradeoffs (see [15,28]), as well as to optimal register allocation for compilers (see [30]). Connections have also been made to pursuit and evasion games and graph searching (see [21,27]). Another version (black pebbling) is used to reorder large sparse matrices to minimize in-core storage during an out-of-core Cholesky factorization scheme (see [12,22,24]). A third version yields results in computational geometry in the rigidity of graphs,

Pebbling in semi-2-trees

“…Moreover, [8] and [4] characterized those graphs having π(G) = n + 1, and it was shown in [10] that one can recognize such graphs in quartic time, improving on the order n 3 m algorithm of [3]. Beginning a program to study for which graphs their pebbling number can be computed in polynomial time, the authors of [1] produced a formula for the family of split graphs that involves several cases. For a given graph, finding to which case it belongs takes O(n 1.41 ) time.…”

“…In opposition to the small diameter, large tree width case of split graphs, we turn here to chordal graphs with large diameter and small tree width 1 . In this paper we study 2-paths, the sub-class of 2-trees whose graphs have exactly two simplicial vertices,…”

Pebbling in 2-Paths

“…[12]) for what families of graphs G the pebbling number π(G) (defined below) can be calculated in polynomial time. One possible family posited in [1] is that of chordal graphs, most likely with some restriction, such as bounded diameter or treewidth, for example. This paper follows a sequence ( [1,2,3,4]) intended to provide at least a partial answer to this line of inquiry, which has led us to make the following conjecture.…”

“…One possible family posited in [1] is that of chordal graphs, most likely with some restriction, such as bounded diameter or treewidth, for example. This paper follows a sequence ( [1,2,3,4]) intended to provide at least a partial answer to this line of inquiry, which has led us to make the following conjecture. We define the pyramid to be the triangulated 6-cycle abcdef with interior triangle bdf , and say that a graph is H-free if it does not contain H as an induced subgraph.…”

“…This case is a key stepping stone toward the graph classes k-paths and interval graphs. At each stage of the sequence of papers [1,2,3,4] mentioned above we have discovered new hurdles that have required new techniques which have expanded our understanding of pebbling in chordal graphs, such as the technical lemmas found in Section 3. The critical pieces of the puzzle in this paper are the new chordal lemmas found in Section 3.3, as well as the careful interplay between the t-wide and t-long cases in the inductive proof of Section 5.…”