Calculating genus polynomials via string operations and matrices

Abstract

To calculate the genus polynomials for a recursively specifiable sequence of graphs, the set of cellular imbeddings in oriented surfaces for each of the graphs is usually partitioned into imbedding-types. The effects of a recursively applied graph operation τ on each imbedding-type are represented by a production matrix. When the operation τ amounts to constructing the next member of the sequence by attaching a copy of a fixed graph H to the previous member, Stahl called the resulting sequence of graphs an H-linear family. We demonstrate herein how representing the imbedding types by strings and the operation τ by string operations enables us to automate the calculation of the production matrices, a task requiring time proportional to the square of the number of imbedding-types.