Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan

TL;DR

This paper introduces the first greedy algorithm for listing all spanning trees of a graph with minimal pivot changes, focusing on the fan graph, and provides efficient ranking/unranking algorithms.

Contribution

It presents a novel greedy method for generating pivot Gray codes for spanning trees, along with efficient algorithms for ranking and unranking, applicable to specific graph classes.

Findings

First greedy algorithm for pivot Gray codes of spanning trees

O(1) amortized time recursive listing algorithm

O(n)-time ranking and unranking algorithms

Abstract

We consider the problem of listing all spanning trees of a graph $G$ such that successive trees differ by pivoting a single edge around a vertex. Such a listing is called a "pivot Gray code", and it has more stringent conditions than known "revolving-door" Gray codes for spanning trees. Most revolving-door algorithms employ a standard edge-deletion/edge-contraction recursive approach which we demonstrate presents natural challenges when requiring the "pivot" property. Our main result is the discovery of a greedy strategy to list the spanning trees of the fan graph in a pivot Gray code order. It is the first greedy algorithm for exhaustively generating spanning trees using such a minimal change operation. The resulting listing is then studied to find a recursive algorithm that produces the same listing in $O(1)$-amortized time using $O(n)$ space. Additionally, we present $O(n)$-time algorithms for ranking and unranking the spanning trees for our listing; an improvement over the generic $O(n^3)$-time algorithm for ranking and unranking spanning trees of an arbitrary graph. Finally, we discuss how our listing can be applied to find a pivot Gray code for the wheel graph.