Search strategies for developing characterizations of graphs without small wheel subdivisions

TL;DR

This paper introduces algorithms and programs to automate the generation and testing of graphs for characterizations related to the Subgraph Homeomorphism Problem, especially for wheel graphs with six and seven spokes.

Contribution

It presents a main algorithm for automating case analysis in proofs of subgraph homeomorphism for wheel graphs, aiding in developing characterizations of graphs without small wheel subdivisions.

Findings

Automated generation and testing of relevant graphs.

Algorithms facilitate proofs for larger wheel graphs.

Potential application to broader graph characterization problems.

Abstract

Practical algorithms for solving the Subgraph Homeomorphism Problem are known for only a few small pattern graphs: among these are the wheel graphs with four, five, six, and seven spokes. The length and difficulty of the proofs leading to these algorithms 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 such analysis is needed for the wheel with seven spokes. This paper describes algorithms and programs used to automate the generation and testing of the graphs that arise as cases in these proofs. The main algorithm given may be useful in a more general context, for developing other characterizations of SHP-related properties.