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.

An Euler‐genus approach to the calculation of the crosscap‐number polynomial

“…Nonetheless, we shall see that the Euler‐genus polynomial can be calculated from a simultaneous recursion of partial Euler‐genus polynomials. From , we know that the number of embeddings types can be quite large. We illustrate in some examples the surprising discovery that the number of embedding types for the Euler‐genus polynomial simultaneous recursion need not exceed the number of types for the genus polynomial recursion.…”

“…Elsewhere we allow a “linear sequence” of copies of the iterated subgraph H to be amalgamated along single edges , or along equivalently embedded copies of arbitrary subgraphs , rather than requiring the amalgamations to be at sets of vertices. The difference is that, as illustrated in Figure below, our present definition of an H ‐linear sequence may leave some “loose ends.” Accordingly, the polynomials of interest for a linear sequence of graphs with loose ends at its extremities may require division by some scalar multiples to eliminate the algebraic contribution of the loose ends.…”

“…Elsewhere we allow a "linear sequence" of copies of the iterated subgraph to be amalgamated along single edges [35], or along equivalently embedded copies of arbitrary subgraphs [16], rather than requiring the amalgamations to be at sets of vertices. The difference is that, as illustrated in Figure 6 below, our present definition of an -linear sequence may leave some "loose ends."…”

Linear Time Canonicalization and Enumeration of Non-Isomorphic 1-Face Embeddings

“…Even for general graphs as small as seven vertices and sparse graphs with around 10 vertices, enumerating topological invariants might require running time in the order of magnitude in hundreds of hours, see [6,9,14,33]. Therefore, our data sets are both challenging and large enough to evaluate the efficiency of the algorithms.…”

“…It is conceivable that the proximity might be reflected by the biological gap representation, which we introduce in Section 3. Therefore, protein or DNA sequences might be more 0 0 3 3 746 107 320 6 158 500 382 165 280 1 240 4 594 836 922 960 7 178 457 399 105 280 2 3 396 5 20 761 712 301 960 8 0 Table 1: The genus distribution of K 7 [14]. The minimum and maximum genus of K 7 is 1 and 7, respectively.…”