Characterizations and algorithms for topological containment of wheel graphs

Topological containment is an important substructure relation in graph theory. A graph G is said to topologically contain another pattern graph H if a graph isomorphic to H can be obtained from G by performing a series of deletions and contractions, where contractions are limited to edges with at least one endvertex of degree 2. It is also said that G contains a subdivision of H. Determining whether some input graph G topologically contains a particular fixed pattern graph is a widely examined problem with many applications. One of the most important is Kuratowski's well known theorem characterizing planarity, which tells us that a graph is non-planar if and only if it topologically contains at least one of the graphs K₅ and K_3,3.

The problem of topological containment has been shown by Robertson and Seymour to be polynomial-time solvable for any fixed pattern graph H. However, practical characterizations and algorithms have only been developed for a few small pattern graphs, among these being the wheels with four and five spokes. This thesis looks at topological containment of wheels with six and seven spokes. The main results are two theorems: one that characterizes graphs with no subdivisions of W₆, and one that gives a characterization (up to bounded size pieces) of graphs with no subdivisions of W₇. These theorems form the basis for good characterizations and efficient algorithms for topological containment of W₆ and W₇. A result is also given strengthening the previous characterization of graphs with no W₅-subdivisions.

We find new types of separating sets such that dividing a graph G along such a separating set will not alter the existence or otherwise of a W_k-subdivision in G (where k = 6 or 7, depending on which of the main theorems forbids the separating set). Similarly, we find new local reductions such that performing such a reduction on G will not alter the existence or otherwise of a W_k-subdivision in G (where k = 6 or 7, again). These separating sets and reductions are used in the two main theorems.

The length and difficulty of the proofs of these theorems increase greatly as the size of the pattern graph increases. Proving a result for the wheel with six spokes requires extensive case analysis on many small graphs, and even more analysis is needed for the wheel with seven spokes. A computer program was written to automate the generation and testing of some of the graphs that arise as cases in these proofs. The main algorithm used in this program may be useful in a more general context, for developing other characterizations relating to topological containment. 

Awards: Winner of the Mollie Holman Doctoral Medal for Excellence, Faculty of Information Technology, 2009.