Genus Distributions of cubic series-parallel graphs

TL;DR

This paper presents a quadratic-time algorithm to compute the genus distribution of certain series-parallel graphs, extending to graphs of treewidth 2 and maximum degree 3, aiding topological graph analysis.

Contribution

It introduces a novel quadratic-time algorithm for genus distribution of specific series-parallel graphs and extends it to broader classes of graphs with bounded treewidth and degree.

Findings

Quadratic-time algorithm for genus distribution of 3-regular, biconnected series-parallel graphs.

Extension of the algorithm to all biconnected series-parallel graphs with max degree 3.

Application to graphs of treewidth 2 and max degree 3 using bar-amalgamation.

Abstract

We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.