Number Of Pebbles Research Articles

Hub Cover Pebbling Number

The hub cover pebbling number, h ∗ ( G ) , of a graph $G$, is the least non-negative integer such that from all distributions of h ∗ ( G ) pebbles over the vertices of G , it is possible to place at least one pebble each on every vertex of a set of vertices of a hub set for G using a sequence of pebbling move operations, each pebbling move operation removes two pebbles from a vertex and places one pebble on an adjacent vertex. Here we compute the hub cover pebbling number for wheel related graphs.

Identification of TRISO pebbles at arbitrary orientation using pairs of X-ray radiographs

TRISO (tri-structural isotropic) fuel is anticipated to be one of the fuel types for future small modular reactors. A TRISO-fuelled pebble in a pebble bed reactor (PBR) consists of thousands of TRISO fuel particles embedded inside a graphite sphere. The diameter of each of the fuel particles is approximately 1 mm. A PBR can contain hundreds of thousands of pebbles, each with a diameter of approximately 6 cm. Unlike in traditional reactor designs, which have a relatively small number of discrete fuel assemblies, the large number of pebbles inside a PBR makes the tracking of individual fuel pebbles difficult. The distribution of TRISO particles within the pebble provides a unique fingerprint that identifies a pebble and can be used to track it. This fingerprint can be determined using a standard X-ray radiograph or a more time-consuming X-ray computed tomography scan. When an individual radiograph is used, the pebble image is compared to a library of reference images of the pebbles. To obtain a good match, the orientation of the pebble must be close to the orientation used for the reference image. As a pebble exits the reactor in a random orientation, the pebble must be reorientated before its identification can take place. This paper discusses the addition of a reference particle in the non-fuelled region of the pebble, as well as the simultaneous acquisition of pairs of radiographs at 90° with respect to each other, to aid pebble reorientation and the development of an identification framework. The framework follows a hybrid approach in which the pebble is first reorientated using machine learning techniques before being identified using traditional computer vision techniques. Other effects such as the impact of transmutation of the elements inside the fuel and the radioactivity of irradiated pebbles are also discussed.

Restricted optimal pebbling is NP-hard

Consider a distribution of pebbles on a graph. A pebbling move removes two pebbles from a vertex and place one at an adjacent vertex. A vertex is reachable under a pebble distribution if it has a pebble after the application of a sequence of pebbling moves. A pebble distribution is solvable if each vertex is reachable under it. The size of a pebble distribution is the total number of pebbles. The optimal pebbling number π∗(G) is the size of the smallest solvable distribution. A t-restricted pebble distribution places at most t pebbles at each vertex. The t-restricted optimal pebbling number πt∗(G) is the size of the smallest solvable t-restricted pebble distribution. We show that deciding whether π2∗(G)≤k is NP-complete. We prove that πt∗(G)=π∗(G) if δ(G)≥2|V(G)|3−1 and we show infinitely many graphs which satisfies δ(H)≈12|V(H)| but πt∗(H)≠π∗(H), where δ denotes the minimum degree.

NDC Pebbling Number for Some Class of Graphs

Let \(G\) be a connected graph. A pebbling move is defined as taking two pebbles from one vertex and the placing one pebble to an adjacent vertex and throwing away the another pebble. A dominating set \(D\) of a graph \(G=(V,E)\) is a non-split dominating set if the induced graph \(\) is connected. The Non-split Domination Cover(NDC) pebbling number, \(\psi_{ns}(G)\), of a graph $G$ is the minimum of pebbles that must be placed on \(V(G)\) such that after a sequence of pebbling moves, the set of vertices with a pebble forms a non-split dominating set of \(G\), regardless of the initial configuration of pebbles. We discuss some basic results and determine \(\psi_{ns}\) for some families of standard graphs.

On the secure vertex cover pebbling number

A new graph invariant called the secure vertex cover pebbling number, which is a combination of two graph invariants, namely, ‘secure vertex cover’ and ‘cover pebbling number’, is introduced in this paper. The secure vertex cover pebbling number of a graph, [Formula: see text], is the minimum number [Formula: see text] so that every distribution of [Formula: see text] pebbles can reach some secure vertex cover of [Formula: see text] by a sequence of pebbling moves. In this paper, the complexity of the secure vertex cover problem and secure vertex cover pebbling problem are discussed. Also, we obtain some basic results and the secure vertex cover pebbling number for complete [Formula: see text]-partite graphs, paths, Friendship graphs, and wheel graphs.

Maximal matching cover pebbling number for variants of hypercube

An edge pebbling move is defined as the removal of two pebbles from one edge and placing one on the adjacent edge. The maximal matching cover pebbling number, fmmcp(G), of a graph G, is the minimum number of pebbles that must be placed on E(G), such that after a sequence of pebbling moves the set of edges with pebbles forms a maximal matching regardless of the initial configuration. In this paper, we find the maximal matching cover pebbling number for variants of hypercube.

Hopping of elliptical pebbles on the beach and laboratory results

On sandy ocean beaches, amongst many irregularly shaped pebbles, a surprisingly large number of pebbles that are nearly perfect ellipsoids may also be found. These elliptical stones are characterized by the density rho of the mineral and by the length of their symmetry axes (a, b, c) with (a > b > c). The grinding process which forms the ab-plane of the ellipsoid relates to the rotation around the axis of the greatest moment of inertia (c-axis) in the surf. In a recent paper by fairly simple physical arguments, it was shown that by this rotation around the c-axis the ellipticity varepsilon always increases. This is in agreement with the empirical observation that on sandy beaches of the oceans no spherical or disc-like pebbles can be found. If the pebble velocity during the rotation becomes too large, then hopping of the pebble sets in. This is caused by the fact that the centre-of-mass acceleration during fast rotation exceeds the gravity acceleration g. Since greater forces occur for jumping than for rolling, the hopping can have a strong influence on the b/a-ratio of the elliptical pebble. In this work, it will be shown that the b-axis is worn away more than the a-axis by erosion with jumping. In a laboratory study with a model of an elliptical cylinder, the theoretical results were checked and partially confirmed.

Monophonic pebbling number of some families of cycles

Let [Formula: see text] be a simple connected graph. A configuration of pebbles is a function from [Formula: see text] to a set of integers. Consider a configuration of pebbles on [Formula: see text]. A pebbling move consists of removing two pebbles off a vertex and placing one on an adjacent vertex. The monophonic pebbling number of [Formula: see text] is the smallest integer [Formula: see text] from which we can put one pebble to a target using monophonic path through pebbling moves. The monophonic [Formula: see text]-pebbling number of [Formula: see text] is the smallest positive integer [Formula: see text] such that from any configuration of [Formula: see text] pebbles we can put [Formula: see text] pebbles on a target using monophonic path. Here we discuss these concepts for families of cycles.

Detour Pebbling Number on Some Commutative Ring Graphs

Detour Pebbling Number on Some Commutative Ring Graphs

Pebbling in powers of paths

The t-fold pebbling number, πt(G), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least t pebbles on any specified vertex via pebbling moves. It has been conjectured that the pebbling numbers of pyramid-free chordal graphs can be calculated in polynomial time.The kth power G(k) of the graph G is obtained from G by adding an edge between any two vertices of distance at most k from each other. The kth power of the path Pn on n vertices is an important class of pyramid-free chordal graphs, and is a stepping stone to the more general class of k-paths and the still more general class of interval graphs. Pachter, Snevily, and Voxman (1995) calculated π(Pn(2)), Kim (2004) calculated π(Pn(3)), and Kim and Kim (2010) calculated π(Pn(4)). In this paper we calculate πt(Pn(k)) for all n, k, and t.For a function D:V(G)→N, the D-pebbling number, π(G,D), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least D(v) pebbles on each vertex v via pebbling moves. It has been conjectured that π(G,D)≤π|D|(G) for all G and D. We make the stronger conjecture that every G and D satisfies π(G,D)≤π|D|(G)−(s(D)−1), where s(D) counts the number of vertices v with D(v)>0. We prove that trees and Pn(k), for all n and k, satisfy the stronger conjecture.The pebbling exponenteπ(G) of a graph G was defined by Pachter et al., to be the minimum k for which π(G(k))=n(G(k)). Of course, eπ(G)≤diam(G), and Czygrinow, Hurlbert, Kierstead, and Trotter (2002) proved that almost all graphs G have eπ(G)=1. Lourdusamy and Mathivanan (2015) proved several results on πt(Cn2), and Hurlbert (2017) proved an asymptotically tight formula for eπ(Cn). Our formula for πt(Pn(k)) allows us to compute eπ(Pn) within additively narrow bounds.

Learning How to Make “Good Enough” Estimations

The ability to estimate physical quantities is a useful skill that can help develop critical thinking and scientific literacy. This lesson provides an accessible way to teach students how to estimate physical quantities by focusing on three key aspects: making assumptions based on previous experiences, explicitly converting units, and communicating the solution clearly. Students are first taught how to perform each of these three steps through a problem-based interactive learning cycle led by the instructor before performing further estimates in groups during a minimally structured investigation. Because this lesson is lean on physics content, it can be used in a variety of classes and is ideal as an early-term laboratory activity. Students engage with the activity by investigating spaces, planning an approach to a solution, and ultimately presenting that solution to one another. <em>Primary Image:</em> Science Courtyard, 2021. This is a picture of the courtyard near the science building. The courtyard is used for estimation problems such as approximating the number of pebbles within or bricks used in the buildings. The author is the copyright holder of this image. &nbsp;

On the pebbling numbers of some snarks

On the pebbling numbers of some snarks

DETOUR PEBBLING ON CARTESIAN PRODUCT GRAPHS

Given a distribution 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 those pebbles on an adjacent vertex. The t - pebbling number of G is the smallest number, ft(G) such that from any distribution of ft(G) pebbles, it is possible to move t pebbles to any specified target vertex by a sequence of pebbling moves. The detour pebbling number of a graph f ∗(G) is the smallest number such that from any distribution of f ∗(G) pebbles, it is possible to move a pebbles to any specified target vertex by a sequence of pebbling moves using a detour path. In this paper, we find the detour pebbling number for some Cartesian product graphs and also the detour t - pebbling number for those cartesian product graphs.

A new lower bound on the optimal pebbling number of the grid

A pebbling move on a graph consists of removing 2 pebbles from a vertex and adding 1 pebble to one of the neighbouring vertices. A vertex is called reachable if we can put 1 pebble on it after a sequence of moves. The optimal pebbling number of a graph is the minimum number p such that there exists a distribution of p pebbles so that each vertex is reachable. For the case of an m×n square grid, Győri, Katona and Papp recently showed that its optimal pebbling number is at least 213nm≈0.1538nm and at most 27nm+O(n+m)≈0.2857nm. We improve the lower bound to 509228593nm+O(m+n)≈0.1781nm.

Comparison of clumps and rigid blocks in three-dimensional DEM simulations: curvature-based shape characterization

Clumps and rigid blocks are two established models in the discrete element method (DEM) for representing digitized shapes of granules. However, few studies compared their shape characterization effectiveness and computational efficiency in describing actual shapes with different accuracies. This research proposes a two-dimensional roundness index (RI2D) and concave index (CI) to quantify surface morphology using a curvature-based method. After collecting material shape information via laser scanning, the models were assessed through three-dimensional DEM simulations at the grain and specimen levels, with balls used as a reference. The findings confirm RI2D and CI as two reliable indicators of the degree to which clumps and rigid blocks vary at the angular scale. Based on RI2D, both clumps and rigid blocks can robustly and reasonably characterize irregular surfaces. However, further improving the accuracy of clumps while satisfying CI requires a significantly greater number of pebbles (spherical units); rigid blocks are incapable of capturing the concavity feature of real particles. RI2D and CI correlate well with the final state in column collapse tests and the shear strength in coupled DEM − FDM (finite difference method) triaxial tests. The difference in simulating granule mechanical behaviors between the two particle models is insignificant with varying levels of accuracy.

Growth after the streaming instability: The radial distance dependence of the planetary growth

Streaming instability is hypothesized to be triggered at particular protoplanetary disk locations where the volume density of the solid particles is enriched comparable to that of the gas. A ring of planetesimals thus forms when this condition is fulfilled locally. These planetesimals collide with each other and accrete inward drifting pebbles from the outer disk to further increase masses. We investigate the growth of the planetesimals that form in a ring-belt at various disk radii. Their initial mass distributions are calculated based on the formula summarized from the streaming instability simulations. We simulate the subsequent dynamical evolution of the planetesimals with a protoplanetary disk model based either on the minimum mass solar nebula (MMSN) or on the Toomre stability criterion. For the MMSN model, both pebble accretion and planetesimal accretion are efficient at a close-in orbit of 0.3 AU, resulting in the emergence of several super-Earth mass planets after 1 Myr. For comparison, only the most massive planetesimals undergo substantial mass growth when they are born at r = 3 AU, while the planetesimals at r = 30 AU experience little or no growth. On the other hand, in the denser Toomre disk, the most massive forming planets can reach Earth mass at t = 1 Myr and reach a mass between that of Neptune and that of Saturn within 3 Myr at 30 AU and 100 AU. Both the pebble and planetesimal accretion rate decrease with disk radial distance. Nevertheless, planetesimal accretion is less pronounced than pebble accretion at more distant disk regions. Taken together, the planets acquire higher masses when the disk has a higher gas density, a higher pebble flux, and/or a lower Stokes number of pebbles.

Secure Domination Cover Pebbling Number of Join of graphs

Objectives: To find the secure domination cover pebbling number for the join of two graphs G(p; q) and G ′ (p ′ ;q ′ ). Methods: We define Secure domination cover pebbling number, fsd p(G), of a graph G as the minimum number of pebbles that must be placed on V(G) such that, after a sequence of pebbling moves, the set of vertices with pebbles forms a secure dominating set for G. Findings: We found the secure domination cover pebbling number for the join of two graphs G(p; q) and Kn. Also, the secure domination cover pebbling number for the join of two graphs G(p; q) and G ′ (p ′ ;q ′ ) is determined when the cardinality of the secure dominating set is 2, 3 and 4. A generalization for the secure domination cover pebbling number of path Pn is also found. Subject Mathematics Classification: 05C38, 05C69 Keywords: Graph pebbling; Secure domination; Cover pebbling number; Secure domination cover pebbling number 2010 Subject Mathematics Classification: 05C38; 05C69

On Secure Total Domination Cover Pebbling Number

In this paper, we introduce a new graph invariant called the secure total domination cover pebbling number, a combination of two graph invariants, namely, `secure total domination' and `cover pebbling number'. The secure total domination cover pebbling number of a graph \(G\), denoted by \(f_{stdp}(G)\), is the minimum number of pebbles that are required to place on \(V(G)\), such that after a sequence of pebbling moves, the set of vertices with pebbles forms a total secure dominating set under any configuration of pebbles to the vertices of graph \(G\). The secure total domination cover pebbling number for join of two graphs \(G(p,q)\) and \(G'(p',q')\) is determined. Also, a generalization of secure total domination cover pebbling number for some families of graphs such as complete graph \(K_n\), complete bipartite graph \(K_{p,q}\), complete $y$-partite graph \(K_{p_1,p_2,\ldots,p_y}\) and path \(P_n\) is found.

Monophonic pebbling number and t-pebbling number of some graphs

Assume G is a graph with some pebbles distributed over its vertices. A pebbling move is when two pebbles are removed from one vertex, one is thrown away, and the other is moved to an adjacent vertex. The monophonic pebbling number, of a connected graph G, is the least positive integer n such that any distribution of n pebbles on G allows one pebble to be carried to any specified but arbitrary vertex using monophonic path by a sequence of pebbling operations. The least positive integer n such that any distribution of n pebbles on G allows t pebbles to be moved to any specified but arbitrary vertex by a sequence of pebbling moves using monophonic path is the monophonic t-pebbling number The monophonic pebbling number and monophonic t-pebbling number of Jahangir graphs, paths and square of paths are determined in this study.

Pebble guided optimal treasure hunt in anonymous graphs

We study the problem of treasure hunt in a graph by a mobile agent. The nodes in the graph are anonymous and the edges at any node v of degree deg(v) are labeled arbitrarily as 0,1,…,deg(v)−1. A mobile agent, starting from a node, must find a stationary object, called treasure that is located on an unknown node at a distance D from its initial position. The agent finds the treasure when it reaches the node where the treasure is present. The time of treasure hunt is defined as the number of edges the agent visits before it finds the treasure. The agent does not have any prior knowledge about the graph or the position of the treasure. An Oracle, that knows the graph, the initial position of the agent, and the position of the treasure, places some pebbles on the nodes, at most one per node, of the graph to guide the agent towards the treasure.We target to answer the question: what is the fastest possible treasure hunt algorithm regardless of the number of pebbles are placed?We show an algorithm that uses O(Dlog⁡Δ) pebbles to find the treasure in a graph G in time O(Dlog⁡Δ), where Δ is the maximum degree of a node in G and D is the distance from the initial position of the agent to the treasure. We show a matching lower bound of Ω(Dlog⁡Δ) on time of the treasure hunt using any number of pebbles.