Searching and sweeping graphs: a brief survey

This papers surveys some of the work done on trying to capture an intruder in a graph. If the intruder may be located only at vertices, the term searching is employed. If the intruder may be located at vertices or along edges, the term sweeping is employed. There are a wide variety of applications for searching and sweeping. Old results, new results and active research directions are discussed

Extremal Results for Peg Solitaire on Graphs

In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families

Cops, robbers and firefighters on graphs

This thesis focuses on the game of cops and robbers on graphs, which was introduced independently by Quilliot in 1978 and by Nowakowski and Winkler in 1983, and one of its variants, the firefighter problem. In the game of cops and robbers, the cops start by choosing their starting positions on vertices of a graph, then the robber chooses his starting point. Then, they move each in turn along the edges of the graph. The basic objective is to determine whether the cops have a strategy which allows them to catch the robber. Looped vertices allow the cops and the robber to pass their turn. The first chapter explores the effect of loops on the cop number and the capture time. It provides examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play. In the second chapter, we investigate the links between this game and algebraic topology. We extend the game of cops and robbers on graphs by considering the case where the cops chase the image of the robber by a graph homomorphism. We prove that the cop number associated with a graph homomorphism is a homotopic invariant. Homotopies between graph homomorphisms or homotopy equivalences between graphs allow us to compare their cop numbers and also their capture times. Finally, using homotopic invariants such as homology groups, we investigate structural properties of copwin graphs. Finally, in the third chapter, we explore the Firefighter problem, introduced by Hartnell in 1995, where a fire spreads through a graph while a player chooses which vertices to protect in order to contain it. While focusing on the case of trees, we also consider a variant game called Fractional Firefighter in which the amount of protection allocated to a vertex lies between 0 and 1. While most of the work in this area deals with a constant amount of firefighters available at each turn, we consider three research questions which arise when including the sequence of firefighters as part of the instance. We first introduce an online version of both Firefighter and Fractional Firefighter, in which the number of firefighters available at each turn is revealed over time. We show that a greedy algorithm on finite trees is 1/2-competitive for both online versions, which generalises a result previously known for special cases of Firefighter. We also show that the optimal competitive ratio of online Firefighter ranges between 1/2 and the inverse of the golden ratio. Next, given two firefighter sequences, we discuss sufficient conditions for the existence of an infinite tree that separates them, in the sense that the fire can be contained with one sequence but not with the other. To this aim, we study a new purely numerical game called targeting game. Finally, we give sufficient conditions for the fire to be contained on infinite trees, expressed as the asymptotic comparison of the number of firefighters and the size of the tree levels

A balanced reading approach for grade one and two English L1 and EAL learners.

Thesis (M.A.)-University of KwaZulu-Natal, Pietermaritzburg, 2003.This study reports on a balanced reading approach (BRA) designed for a multi-cultural classroom, including both Ll ( first language) and EAL ( English additional language ) Foundation phase learners. The purpose of this study is to explore how interactive reading approaches develop literacy skills for six African learners. The teacher as researcher developed a theoretical model, which informed her pedagogic practices in the balanced reading programme. She also engaged in action research to gain an insight into what teaching approaches, methodologies and resources make EAL learners learn more effectively. This information was disseminated to other educator colleagues. This study also examines parents' views on the BRA and their perceptions of the reading process. Data was drawn from the following sources: teacher observations and interactions with learners, semi-structured interviews with parents of learners, analysis of learner assessment and parental questionnaires. It was concluded that a balanced reading approach which values mother tongue instruction in a supportive learning environment enhances the self concepts and cognitive growth of EAL learners. This study has also demonstrated that collaborative active learning, extensive independent reading, language experience approach, home support , community support, high levels of intrinsic learner motivation and high teacher expectations of learners can positively impact on the EAL learners' academic progress and social growth at school. Keywords: Foundation phase English Additional Language Balanced Reading Approach III Additive Bilingualism Communicative Language Teaching Whole Languag

Individualizing Instruction Through the Use of Classroom Groups

The purpose of this study is twofold: first, to identify those problems classroom teachers meet when organizing groups within the classroom to provide for individual differences in rates of learning, instructional and/or personal needs, and interests; second, to develop some generalizations which are supported by the findings of educational research and experience, and which will give educators some guidance in the area of grouping for instruction

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Bonded: Legacies of Captivity and Fugitivity from Enslavement to Incarceration in the Cape

The contemporary hyper-incarceration of ‘Coloured' South Africans is re-situated within the broader historical dialectics of racialisation and creolisation, traversing from colonial slavery to the modern prison regime. This study uses theorisations of marronage, fugitivity, and hauntology to posit novel understandings of the links between runaway slaves (‘droster1 gangs') and the contemporary ‘Coloured' criminal figure. This dissertation approaches the latter as engaged in traditions of opacity-making, initiated by the former as a production of complex structures of density and unknowability against the epistemic violence of the colonial gaze that seeks to ‘discover', categorise and control. As such, this study proposes to understand collectives of fugitives beyond the lexicons of criminality, on the one extreme, and resistance, on the other. Applying emerging qualitative and arts-based methods, it further offers an innovative methodological framework to strategically listen for the poetics and sonicity of fugitive narratives, highlighting the incondensable movements therein of dense temporalities, opacities, and personal and collective narration. Specifically, through a poetry- and performance-based workshop series, this study collaborates with formerly-incarcerated men to engage with the Cape's history of slavery and marronage, exploring the meanings and relevance of this history through creative writings, group discussions, and performance