A Pivot Gray Code Listing for the Spanning Trees of the Fan Graph

TL;DR

This paper introduces a greedy algorithm for listing all spanning trees of fan graphs with minimal edge changes, along with efficient ranking and unranking algorithms, advancing enumeration methods in graph theory.

Contribution

It presents the first greedy algorithm for exhaustively generating spanning trees with single-edge pivot operations and provides efficient ranking/unranking algorithms.

Findings

First greedy listing of spanning trees with pivot operations

O(1) amortized time recursive listing algorithm

O(n) ranking and unranking algorithms

Abstract

We use a greedy strategy to list the spanning trees of the fan graph, $F_n$, such that successive trees differ by pivoting a single edge around a vertex. 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.