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Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

THOMAS, DANIEL,JAMES,RHYS (2020) Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs. Doctoral thesis, Durham University.

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In this thesis we investigate three different aspects of graph theory.
Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs.

Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber.

Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses.

Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Faculty and Department:Faculty of Science > Computer Science, Department of
Thesis Date:2020
Copyright:Copyright of this thesis is held by the author
Deposited On:05 Aug 2020 11:45

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