The mincut graph of a graph

In this paper we introduce an intersection graph of a graph $G$, with vertex set the minimum edge-cuts of $G$. We find the minimum cut-set graphs of some well-known families of graphs and show that every graph is a minimum cut-set graph, henceforth called a \emph{mincut graph}. Furthermore, we show that non-isomorphic graphs can have isomorphic mincut graphs and ask the question whether there are sufficient conditions for two graphs to have isomorphic mincut graphs. We introduce the $r$-intersection number of a graph $G$, the smallest number of elements we need in $S$ in order to have a family $F=\{S_1, S_2 \ldots , S_i\}$ of subsets, such that $|S_i|=r$ for each subset. Finally we investigate the effect of certain graph operations on the mincut graphs of some families of graphs

Graphs and graph polynomials

A dissertation submitted to the School of Mathematics in fulfilment of the requirements for the degree of Master of Science School of Mathematics University of the Witwatersrand, October 2017In this work we study the k-defect polynomials of a graph G. The k defect polynomial is a function in λ that gives the number of improper colourings of a graph using λ colours. The k-defect polynomials generate the bad colouring polynomial which is equivalent to the Tutte polynomial, hence their importance in a more general graph theoretic setting. By setting up a one-to-one correspondence between triangular numbers and complete graphs, we use number theoretical methods to study certain characteristics of the k-defect polynomials of complete graphs. Specifically we are able to generate an expression for any k-defect polynomial of a complete graph, determine integer intervals for k on which the k-defect polynomials for complete graphs are equal to zero and also determine a formula to calculate the minimum number of k-defect polynomials that are equal to zero for any complete graph.XL201

A tacit component to acquiring critical thinking skills? : analysing expert knowledge in order to enhance the efficiency of learning critical thinking skills

M.A.The rationale for this study is given. The rational, as explicit reasoning and knowledge, is set against the tacit dimension, and some of the implications of a tacit dimension to critical thinking skills are mentioned: Tacit knowledge: The idea of the tacit dimension is introduced as an area traditionally neglected and ignored by western philosophy and the scientific world-view, even though it has had a number of prominent advocates over the past few centuries. Two views of knowledge are identified - one emphasising a detached rational stance, the other emphasising the importance of experience in thinking and knowledge. A thought experiment: Creating a Critical Thinking Expert System: In building an expert system it is required to supply a knowledge base, an inference engine, and a user interface. It is concluded from the experiment that critical thinking cannot be mastered by merely knowing the definitions and rules - it is a skill that develops with practice. Tacit knowledge is temporarily defined as that "something extra" which the expert has, but is unable to give explicitly in his/her instructions without showing us how to do it. The expert's non-rule-following behaviour: Dreyfus and Dreyfus's five stages of becoming an expert are outlined. It is argued that, rather than being able to apply rules really quickly, the expert actually acts without following the rules. Experts simply do what normally works. The question is posed whether it is reasonable to compare the skill of the master craftsman with that of the expert thinker