A 2-Approximation for the Height of Maximal Outerplanar Graph Drawings

In this thesis, we study drawings of maximal outerplanar graphs that place vertices on integer coordinates. We introduce a new class of graphs, called umbrellas, and a new method of splitting maximal outerplanar graphs into systems of umbrellas. By doing so, we generate a new graph parameter, called the umbrella depth (ud), that can be used to approximate the optimal height of a drawing of a maximal outerplanar graph. We show that for any maximal outerplanar graph G, we can create a flat visibility representation of G with height at most 2ud(G) + 1. This drawing can be transformed into a straight-line drawing of the same height. We then prove that the height of any drawing of G is at least ud(G) + 1, which makes our result a 2-approximation for the optimal height. The best previously known approximation algorithm gave a 4-approximation. In addition, we provide an algorithm for finding the umbrella depth of G in linear time. Lastly, we compare the umbrella depth to other graph parameters such as the pathwidth and the rooted pathwidth, which have been used in the past for outerplanar graph drawing algorithms

Validation of the Postnatal Perceived Stress Inventory in a French-speaking population of primiparous women

Objective To develop a Postnatal Perceived Stress Inventory (PNPSI) and assess its psychometric properties. Design Cross-sectional quantitative study. Setting One nurse-managed labor and delivery unit in a university hospital in a major metropolitan area. Participants One hundred seventy-nine (179) primiparous French speaking women who gave birth at term. Methods The PNPSI was validated at 6 weeks postpartum. Its predictive validity for depression and anxiety was assessed at the same time. Results The exploratory analysis revealed a 19-item structure divided into six factors. This inventory has good internal consistency (Cronbach's alpha = .815). The predictive validity shows that the PNPSI significantly predicts depression and anxiety at 6 weeks postpartum, and that certain factors are particularly prominent. Conclusion The PNPSI's psychometric properties make it a useful tool for future research to evaluate interventions for perceived stress during the postnatal period. Its predictive power for depression indicates that it is also a promising tool for clinical settings