On Pebbling Graphs by Their Blocks

Curtis, Dawn; Hines, Taylor; Hurlbert, Glenn; Moyer, Tatiana

doi:10.1515/integ.2009.033

Cited by 7 publications

(9 citation statements)

References 8 publications

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“…We now outline the remainder of the paper. The main result is Theorem 2.7, which provides the best possible upper bound π t (G) ≤ π(G) + 4t − 4 when G has diameter 2, and proves a conjecture of [7] as a corollary. Furthermore, we obtain an algorithm that places t pebbles on any root from a distribution of π(G) + 4t − 4 pebbles on the n vertices of such G, and that runs in at most 6n + min{3t, m} steps, where G has m edges.…”

Section: Introductionmentioning

confidence: 72%

t-Pebbling and Extensions

Herscovici

Hester

Hurlbert

2012

Graphs and Combinatorics

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Graph pebbling is the study of moving discrete pebbles from certain initial distributions on the vertices of a graph to various target distributions via pebbling moves. A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbors (losing the other as a toll).For t ≥ 1 the t-pebbling number of a graph is the minimum number of pebbles necessary so that from any initial distribution of them it is possible to move t pebbles to any vertex.We provide the best possible upper bound on the t-pebbling number of a diameter two graph, proving a conjecture of Curtis, et al., in the process. We also give a linear time (in the number of edges) algorithm to t-pebble such graphs, as well as a quartic time (in the number of vertices) algorithm to compute the pebbling number of such graphs, improving the best known result of Bekmetjev and Cusack.Furthermore, we show that, for complete graphs, cycles, trees, and cubes, we can allow the target to be any distribution of t pebbles without increasing the corresponding t-pebbling numbers; we conjecture that this behavior holds for all graphs.Finally, we explore fractional and optimal fractional versions of pebbling, proving the fractional pebbling number conjecture of Hurlbert and using linear optimization to reveal results on the optimal fractional pebbling number of vertex-transitive graphs.

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Section: Introductionmentioning

confidence: 72%

t-Pebbling and Extensions

Herscovici

Hester

Hurlbert

2012

Graphs and Combinatorics

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“…Clarke, Hochberg, and Hurlbert in [6] classified graphs of diameter two whose pebbling number is n + 1. Curtis et al [7] proved that π m (n, 2) ≤ n + 7m − 6 and conjectured that π m (n, 2) ≤ n + 4m − 3, which was recently proved by Herscovici et al [11].…”

Section: Introductionmentioning

confidence: 90%

“…Curtis et al. proved that

π_{m} (n, 2) \leq n + 7 m - 6

and conjectured that

π_{m} (n, 2) \leq n + 4 m - 3

, which was recently proved by Herscovici et al. .…”

Section: Introductionmentioning

confidence: 91%

Pebbling Graphs of Fixed Diameter

Postle

2013

Journal of Graph Theory

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Given a configuration of indistinguishable pebbles on the vertices of a connected graph G on n vertices, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one pebble on an adjacent vertex. The m‐pebbling number of a graph G, πm(G), is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least m pebbles on v. When m=1, it is simply called the pebbling number of a graph. We prove that if G is a graph of diameter d and k,m≥1 are integers, then πm(G)≤f(k)n+2k+dm+(2k(2d−1)−f(k))domk(G), where domk(G) denotes the size of the smallest distance k dominating set, that is the smallest subset of vertices such that every vertex is at most distance k from the set, and, f(k)=(2k−1)/k. This generalizes the work of Chan and Godbole (Discrete Math 208 (2008), 15–23) who proved this formula for k=m=1. As a corollary, we prove that πm(G)≤f(⌈d/2⌉)n+O(m+nprefixlnn). Furthermore, we prove that if d is odd, then πm(G)≤f(⌈d/2⌉)n+O(m), which in the case of m=1 answers for odd d, up to a constant additive factor, a question of Bukh (J Graph Theory 52 (2006), 353–357) about the best possible bound on the pebbling number of a graph with respect to its diameter.

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“…Graphs satisfying π(G) = n are called Class 0 and are a topic of much interest (e.g. [2,3,5,6,9,10]). Surveys on the topic can be found in [16,17,19], and include variations on the theme such as k-pebbling, fractional pebbling, optimal pebbling, cover pebbling, and pebbling thresholds, as well as applications to combinatorial number theory and combinatorial group theory (see references).…”

Section: Introductionmentioning

confidence: 99%

“…Here we begin to study for which graphs their pebbling numbers can be calculated in polynomial time. Aiming for tree-like structures (as was considered in [6]), one might consider chordal graphs of various sorts. Moving away from diameter 2, one might consider diameter 3 graphs; recently ( [29]), the tight upper bound of ⌊3n/2⌋ + 2 has been shown for this class.…”

Section: Introductionmentioning

confidence: 99%

Pebbling in Split Graphs

Alcón¹,

Gutiérrez²,

Hurlbert³

2014

SIAM J. Discrete Math.

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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,

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On Pebbling Graphs by Their Blocks

Cited by 7 publications

References 8 publications

t-Pebbling and Extensions

t-Pebbling and Extensions

Pebbling Graphs of Fixed Diameter

Pebbling in Split Graphs

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