2012 Theses Doctoral

Methods for Computing Genus Distribution Using Double-Rooted Graphs

Khan, Imran Farid

This thesis develops general methods for computing the genus distribution of various types of graph families, using the concept of double-rooted graphs, which are defined to be graphs with two vertices designated as roots (the methods developed in this dissertation are limited to the cases where one of the two roots is restricted to be of valence two). I define partials and productions, and I use these as follows: (i) to compute the genus distribution of a graph obtained through the vertex amalgamation of a double-rooted graph with a single-rooted graph, and to show how these can be used to obtain recurrences for the genus distribution of iteratively growing infinite graph families. (ii) to compute the genus distribution of a graph obtained (a) through the operation of self-vertex-amalgamation on a double-rooted graph, and (b) through the operation of edge-addition on a double-rooted graph, and finally (iii) to develop a method to compute the recurrences for the genus distribution of the graph family generated by the Cartesian product of P3 and Pn.