Topics Covered

The Following Three Lectures will each be Delivered in the First 3 Days of the Workshop.

Aspects of distances in graphs and digraphs

By Professor Peter Dankelmann, University of Johannesburg, South Africa

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In this series of lectures, we present several old and new results on distances in graphs and digraphs. We consider four of the most important distance parameters, diameter, radius, Wiener index and average eccentricity.

There are many results on these parameters for undirected graphs in the literature. Results on these parameters in directed graphs are few and far between, and many results that hold for graphs are known not to hold for digraphs in general. However, some of these results have been shown to hold for a large subclass of digraphs that conatains all graphs, Euleriandigraphs.

We present a selection of basic and less basic results on distance parameters in graphs, discuss corresponding results, or the absence thereof, for directed graphs, and discuss these results in the context of Eulerian digraphs. For many of these results we present proofs, demonstrating important techniques in the research on distances.

An Introduction to Domination in Graphs

By Professor Michael Henning, University of Johannesburg, South Africa

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cid:181811316*[email protected]

We present of series of four introductory talks on domination in graphs. We prove elementary lower and upper bounds on the domination number. We discuss Vizing's theorem that bounds the size of a graph in terms of its order and domination number. We show how the domination number can be used to solve the so-called art gallery problem. We present best known bounds to date on the domination number of a graph with given minimum degree in terms of the order of the graph, and show how these bounds can be improved with certain girth conditions. We also discuss best known bounds to date on the total domination number of a graph with given minimum degree in terms of the order of the graph, and show how the powerful tool of transversal in hypergraphs can be used to obtain results on total domination in graphs.

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\bibitem{HaHeHe-20} T. W. Haynes, S. T. Hedetniemi, and M. A. Henning (eds), \emph{Topics in Domination in Graphs}. Series: Developments in Mathematics, Vol. 64, Springer, Cham, 2020. 545 pp.

\bibitem{HaHeHe-21} T. W. Haynes, S. T. Hedetniemi, and M. A. Henning (eds), \emph{Structures of Domination in Graphs.} Series: Developments in Mathematics, Vol. 66, Springer, Cham, 2021. 536 pp.

\bibitem{HaHeHe-23} T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, \emph{Domination in Graphs: Core Concepts} Series: Springer Monographs in Mathematics, Springer, Cham, 2023, 710 pp.

\bibitem{HeVu-book} M. A. Henning and J. H. van Vuuren, Graph and Network Theory: An Applied Approach using Mathematica. \emph{Springer Optimization and its Applications}, 193. Springer, Cham, 2022. XXIX, ISBN: 978-3-031-03856-3; 978-3-031-03857-0. 766 pp.

\bibitem{HeYe-book} M. A. Henning and A. Yeo, \emph{Total domination in graphs.} \emph{Springer Monographs in Mathematics}. Springer, New York, 2013. 178 pp.

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