Graphs with no 7-wheel subdivision

The topological containment problem, TC(H), has been shown to be polynomial-time solvable for any fixed pattern graph H, but practical algorithms have been developed only for a few specific pattern graphs. Among these are the wheels with four, five, and six spokes. This paper examines the topological containment problem where the pattern graph is a wheel with seven spokes, and gives a result that describes graphs with no W7-subdivision, showing how they can be built up, using certain operations, from smaller ‘pieces’ that meet certain conditions. We also discuss algorithmic aspects of the problem.