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Pebbling in Split Graphs

Description

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

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 over 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[superscript β]) time, where β = 2ω/(ω + 1) [= over ∼] 1.41 and ω [= over ∼] 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.

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Date Created
2013-11-30

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Does Diet Quality Mediate the Relationship Between Food Insecurity and Obesity Among Preschoolers?

Description

Food insecurity and childhood obesity are both major public health concerns in the United States of America. Research has not found a definite relationship between childhood obesity and food insecurity to date, with conflicting results being found due to differences

Food insecurity and childhood obesity are both major public health concerns in the United States of America. Research has not found a definite relationship between childhood obesity and food insecurity to date, with conflicting results being found due to differences in sample sizes and protocol for measuring key variables. Preschoolers (children aged 2-5 years) are a population of particular interest as there tends to be improved health behaviors and greater adaptability to change at this period of growth and development. This study aims to evaluate if there is a relationship between food insecurity and childhood obesity with diet quality as a mediator among preschoolers in the Phoenix area. A secondary data analysis from participants (n=154) from the SAGE (Sustainability via Active Garden Education) research project was used to evaluate food insecurity status, diet quality components (kcal, saturated fat, added sugars, and servings of juice, fruits, and vegetables), and anthropometrics (waist circumference and BMI percentile). No significant associations between food insecurity status, diet quality components, and anthropometric data were found. There was an increased rate of food insecurity and childhood overweight/obesity in this sample compared to state and national averages. Further research of high quality is necessary to determine whether a relationship exists between childhood obesity and food insecurity exists and in what context. Additionally, practice and policy will need to be implemented to decrease rates of food insecurity and childhood obesity among Phoenix preschoolers.

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Created

Date Created
2020