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.
Investigation one revealed a moderate-to-large effect size for school-based interventions (n=10) increasing CVF (g=0.75; 95%CI [0.40-1.11]). Multi-level interventions (g=.79 [0.34-1.25]) were more effective than interventions focused on the individual (g=0.67 [0.12-1.22]). In investigations two and three children (78.3% Hispanic; mean ± SD age 53.2±4.5 months) completed a mean ± SD 3.7±2.3 PACER laps and 19.0±5.5 CSMP criteria. Individual and family factors associated with PACER laps included child sex (B=-0.96, p=0.03) and age (B=0.17, p<0.01), parents’ promotion of inactivity (B=0.66, p=0.08) and screen time (B=0.65, p=0.05), and parents’ concern for child’s safety during physical activity (B=-0.36, p=0.09). Child age (B=0.47, p<0.01) and parent employment (B=2.29, p=0.07) were associated with CMSP criteria. At the ECEC level, policy environment quality (B=-0.17; p=0.01) was significantly associated with number of PACER laps completed. Outdoor play environment quality (B=0.18; p=0.03), outdoor play equipment total (B=0.32; p<0.01) and screen time environment quality (B=0.60; p=0.02) were significantly associated with CMSP criteria. Researchers, ECEC teachers and policy makers should promote positive environmental changes to preschool-aged children’s family and ECEC environments, as these environments have the potential to improve CVF and GLS more than programs focused on the child alone.
In Chapter 2, walking for AT was found to be related to smaller waist circumference, lower blood pressure, and lower prevalence of abdominal obesity and hypertension, and that differences may exist based on sex. Walking for AT was not clearly defined, and criteria used to determine the presence of cardiometabolic outcomes were inconsistent. No significant relationships between AT and cardiometabolic health were found in Chapter 3 or 4; however, AT users had slightly better cardiometabolic health. AT users had significantly higher levels of self-reported total physical activity compared to those who did not use AT in Chapter 3. Furthermore, a significant relationship was found between MVPA and diastolic blood pressure. Associations differed by ethnicity, with MVPA being inversely related to body fat in both AA and HL women, but to body mass index only in AA women. AT users were found to be seven times more likely to meet 2018 national MVPA recommendations than non-AT users in Chapter 4. Across all studies, measures of AT were subjective and of low quality, potentially limiting the ability to detect significant findings.
High quality randomized controlled studies should be conducted using clearly defined, objective measures of AT, and analyzed based on sex and race/ethnicity. Clinicians should recommend AT use to promote meeting MVPA recommendations where appropriate, potentially resulting in improved cardiometabolic health. Policymakers should advocate for changes to the built environment to encourage AT use and MVPA to improve public health.
Methods: Survey design and ethnographic field methods were used to develop, test, and validate the Street Food Stand Assessment Tool (SFSAT). Geographic information system and ground-truthing methods were used to identify a sample of street segments across 20 neighborhoods representing low-, middle- and high-income neighborhoods in Mexico City on which to assess the availability, density, variety, and distribution of SFS and the foods and beverages sold at these food venues using the SFSAT.
Results: A sample of 391 SFS were assessed across 791 street segments. Results showed that SFS were found in all neighborhoods. Contrary to the initial hypothesis, most SFS were found in middle-income neighborhoods. While the availability of street foods and beverages was higher in middle-income neighborhoods, the variety was less consistent: fruit/vegetable variety was high in high-income neighborhoods whereas processed snack variety was higher in low-income neighborhoods. SFS were most often distributed near homes, transportation centers, and worksites across the three neighborhood income levels.
Conclusion: This study bridged the gap in knowledge about the availability, density, variety, and distribution of SFS and products sold at these sources of food by using an assessment tool that was developed, tested, and validated specifically for SFS. The findings showed that SFS were found across all neighborhoods. Furthermore, results also suggested that SFS can be a source of healthy food items. Additional studies are needed to understand the relationship between SFS availability, food consumption, and health outcomes in the Mexican population.