7 research outputs found
Structure and properties of maximal outerplanar graphs.
Outerplanar graphs are planar graphs that have a plane embedding in which each vertex lies on the boundary of the exterior region. An outerplanar graph is maximal outerplanar if the graph obtained by adding an edge is not outerplanar. Maximal outerplanar graphs are also known as triangulations of polygons. The spine of a maximal outerplanar graph G is the dual graph of G without the vertex that corresponds to the exterior region. In this thesis we study metric properties involving geodesic intervals, geodetic sets, Steiner sets, different concepts of boundary, and also relationships between the independence numbers and domination numbers of maximal outerplanar graphs and their spines. In Chapter 2 we find an extension of a result by Beyer, et al. [3] that deals with Hamiltonian degree sequences in maximal outerplanar graphs. In Chapters 3 and 4 we give sharp bounds relating the independence number and domination number, respectively, of a maximal outerplanar graph to those of its spine. In Chapter 5 we discuss the boundary, contour, eccentricity, periphery, and extreme set of a graph. We give a characterization of the boundary of maximal outerplanar graphs that involves the degrees of vertices. We find properties that characterize the contour of a maximal outerplanar graph. The other main result of this chapter gives characterizations of graphs induced by the contour and by the periphery of a maximal outerplanar graph. In Chapter 6 we show that the generalized intervals in a maximal outerplanar graph are convex. We use this result to characterize geodetic sets in maximal outerplanar graphs. We show that every Steiner set in a maximal outerplanar graph is a geodetic set and also show some differences between these types of sets. We present sharp bounds for geodetic numbers and Steiner numbers of maximal outerplanar graphs
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The impact of emotions on student participation in an assessed, online, collaborative activity
There is growing recognition of the importance of emotions in academic online learning contexts. However, there is still little known about the role of emotions in social and collaborative online learning settings, especially the relationship between emotions and student participation. To explore this relationship, this study used a prospective longitudinal research design to follow 46 distance learning students throughout a 3-week assessed, online, collaborative activity. This approach allowed the fluctuating and dynamic aspects of emotions to be explored as well as the relationship between emotions and student participation in the collaborative activity. Self-report data were gathered using a semistructured online diary at five time points throughout the task (once at the start of the collaborative activity, three times during the activity, and the final entry after the activity had finished). Findings revealed that learners generally perceived pleasant emotions (such as relief, satisfaction and enjoyment) to have positive impacts, or no impact, on participation, whereas unpleasant emotions (such as anxiety, frustration, and disappointment) were generally perceived to have negative impacts, or no impact, on participation. Interestingly, however, anxiety, and to a smaller extent frustration, were perceived by a number of students to have positive impacts during the activity. To conclude this paper, implications for educators are highlighted