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

Ben Cameron; Aaron Grubb; Joe Sawada

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arXiv · 2202.01746

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

Ben Cameron·Aaron Grubb·Joe Sawada

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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.

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Ben Cameron, Aaron Grubb, Joe Sawada. 2022-02-03. Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan. https://arxiv.org/abs/2202.01746

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