Characterisation and Hamiltonicity of K_{1,1,t}-minor-free Graphs: A Fan-based Approach

Abstract

This dissertation presents some new results in certain areas of structural graph theory. In particular we are concerned with graph minors, and classes of graphs characterised in part by forbidding certain minors. There are many important results on classes of minor-free graphs, for example Wagner's Theorem, which states that a graph is planar if and only if it does not contain K_5 or K_{3,3} as a minor.

We work specifically with classes of graphs that do not have a complete multipartite graph K_{1,1,t} as a minor. We introduce a type of induced subgraph called a fan and show that several graph properties are preserved under operations with these fans, allowing us to inductively prove significant results for classes of 3-connected K_{1,1,t}-minor-free graphs.

Our first result is a complete structural characterisation of 3-connected K_{1,1,4}-minor-free graphs. We also prove counting results for these, and characterise those that are planar and those that are Hamiltonian.

Secondly, we prove a Hamiltonicity result for the class of 3-connected planar K_{1,1,5}-minor-free graphs. In particular, we prove that with one exception, every 3-connected planar K_{1,1,5}-minor-free graph is Hamiltonian. The exception is the well-known Herschel graph, a bipartite graph on eleven vertices.