Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism

TL;DR

This paper develops algorithms to efficiently generate all non-equivalent spanning trees of series-parallel graphs considering their symmetries, with different approaches based on how the graph's terminals are distinguished.

Contribution

It introduces the first output-linear algorithms for generating non-equivalent spanning trees of series-parallel graphs under various terminal distinctions, without explicitly computing automorphism groups.

Findings

Algorithms operate in output-linear time for oriented graphs.

Extension of algorithms to semioriented graphs with two distinguished terminals.

Discussion of open problems for unoriented series-parallel graphs.

Abstract

In this paper, we investigate the problem of generating the spanning trees of a graph $G$ up to the automorphisms or "symmetries" of $G$. After introducing and surveying this problem for general input graphs, we present algorithms that fully solve the case of series-parallel graphs, under two standard definitions. We first show how to generate the nonequivalent spanning trees of a oriented series-parallel graph $G$ in output-linear time, where both terminals of $G$ have been individually distinguished (i.e. applying an automorphism that exchanges the terminals produces a different series-parallel graph). Subsequently, we show how to adapt these oriented algorithms to the case of semioriented series-parallel graphs, where we still have a set of two distinguished terminals but neither has been designated as a source or sink. Finally, we discuss the case of unoriented series-parallel graphs, where no terminals have been distinguished and present a few observations and open questions relating to them. The algorithms we present generate the nonequivalent spanning trees of $G$ but never explicitly compute the automorphism group of $G$, revealing how the recursive structure of $G$'s automorphism group mirrors that of its spanning trees.