Strong product of graphs: Geodetic and hull number and boundary-type sets

Preprin

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

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

On the geodeticity of the contour of a graph

The eccentricity of a vertex vv in a graph GG is the maximum distance of vv from any other vertex of GG and vv is a contour vertex of GG if each vertex adjacent to vv has eccentricity not greater than the eccentricity of vv. The set of contour vertices of GG is geodetic if every vertex of GG lies on a shortest path between a pair of contour vertices. In this paper, we firstly investigate the existence of operations on graphs that allow to construct graphs in which the contour is geodetic. Then, after providing an alternative proof of the fact that the contour is geodetic in every HHD-free graph, we show that the contour is geodetic in every cactus and in every graph whose blocks are HHD-free or cycles or cographs. Finally, we generalize the above result by introducing the concept of geodetic-contour-preserving class of graphs and by proving that, if each block BB in a graph GG belongs to a class GBGB of graphs which is geodetic-contour-preserving, then the contour of GG is geodetic

On geodetic sets formed by boundary vertices

Let G be a finite simple connected graph. A vertex v is a boundary vertex of G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices. Before showing that one of the main results in [3] does not hold for one of the cases, we establish a realization theorem that not only corrects the mentioned wrong statement but also improves it. Given S ⊆ V (G), its geodetic closure I[S] is the set of all vertices lying on some shortest path joining two vertices of S. We prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, I[∂(G)] = V (G). A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than v. We present some sufficient conditions to guarantee the geodeticity of either the contour Ct(G) or its geodetic closure I[Ct(G)]

On geodetic sets formed by boundary vertices

Let G be a finite simple connected graph. A vertex v is a boundary vertex of G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices. Before showing that one of the main results in [3] does not hold for one of the cases, we establish a realization theorem that not only corrects the mentioned wrong statement but also improves it. Given S ⊆ V (G), its geodetic closure I[S] is the set of all vertices lying on some shortest path joining two vertices of S. We prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, I[∂(G)] = V (G). A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than v. We present some sufficient conditions to guarantee the geodeticity of either the contour Ct(G) or its geodetic closure I[Ct(G)]