Pebbling Graphs of Fixed Diameter

Postle, Luke

doi:10.1002/jgt.21736

Cited by 5 publications

(5 citation statements)

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“…As a way to illustrate the techniques that are used to prove an asymptotic bound on the pebbling number for graphs of diameter four, we improve the

O (\sqrt{n})

term of Bukh's result to

O (1)

. The general bound we obtain has been recently improved by Postle [], but we include it here as it briefly illustrates our technique for the diameter four case, which still is better than Postle's bound. Recall that a vertex is heavy if

p (v) \geq 2^{⌈ \frac{d}{2} ⌉}

, where d is the diameter of the graph.…”

Section: Asymptotic Resultsmentioning

confidence: 99%

Pebbling Graphs of Diameter Three and Four

Postle

Streib

Yerger

2012

Journal of Graph Theory

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Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex and the placement of one of these on an adjacent vertex. The pebbling number of a graph 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 one pebble on v. We improve on a bound of Bukh by showing that the pebbling number of a graph of diameter three on n vertices is at most ⌊3n/2⌋+2, and this bound is best possible. Further, we obtain an asymptotic bound of 3n/2+Θ(1) for the pebbling number of graphs of diameter four. Finally, we prove an asymptotic bound for pebbling graphs of arbitrary diameter, namely that the pebbling number for a diameter d graph on n vertices is at most (2⌈d2⌉−1)n+C′(d), where C′(d) is a constant depending upon d. This also improves another bound of Bukh.

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“…As a way to illustrate the techniques that are used to prove an asymptotic bound on the pebbling number for graphs of diameter four, we improve the

O (\sqrt{n})

term of Bukh's result to

O (1)

p (v) \geq 2^{⌈ \frac{d}{2} ⌉}

, where d is the diameter of the graph.…”

Section: Asymptotic Resultsmentioning

confidence: 99%

Pebbling Graphs of Diameter Three and Four

Postle

Streib

Yerger

2012

Journal of Graph Theory

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“…This completes the proof. ✷ showing that the pebbling bound for diameter 3 graphs given in [29] is tight.…”

Section: For Everymentioning

confidence: 98%

“…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. Combining these two thoughts we study split graphs in this paper, and find that their pebbling numbers can be calculated quickly, in fact, in O(n 1.41 ) time.…”

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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“…Theorem.2.2. In (7), Postle [9]showed that if diameter d of a connected graph with n vertices is odd, then…”

Section: Lower and Upper Bounds Of Pebbling Number Of Watkins Snarkmentioning

confidence: 99%

Pebbling In Watkins Snark Graph

S¹,

Anilkumar²

2019

IJRAT

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The pebbling theory is a study of a mathematical game that is played over a graph. In this game, a pebbling movement means removing two pebbles from a given vertex and adding one pebble to one of its neighbors and removing the other pebble from the game. The pebbling number of a graph G is defined a smallest positive integer required to add a pebble at any target vertex of the graph. It is denoted as . Every vertex of the graph is pebbled irrespective of the initial pattern of pebbles. Cubic graph is also called a 3-regular graph which is used in a real time scenario. In this paper, we have determined the pebbling number of Watkins Snark by constructing a Watkins Flower Snark of vertices, edges, cycles and disjoint sets which are present in the Watkins Snark. It is a connected graph in which bridgeless cubic index is equal to 4 with 75 edges and 50 vertices.

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Pebbling Graphs of Fixed Diameter

Cited by 5 publications

References 12 publications

Pebbling Graphs of Diameter Three and Four

Pebbling Graphs of Diameter Three and Four

Pebbling in Split Graphs

Pebbling In Watkins Snark Graph

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