Vol. 12, No. 4, 2019

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Graphs with at most two trees in a forest-building process

Steve Butler, Misa Hamanaka and Marie Hardt

Vol. 12 (2019), No. 4, 659–670
DOI: 10.2140/involve.2019.12.659
Abstract

Given a graph, we can form a spanning forest by first sorting the edges in a random order, and then only keeping edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with k trees.

We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are nonisomorphic but produce the same probabilities.

Keywords
forests, edge ordering, components, probability
Mathematical Subject Classification 2010
Primary: 05C05
Milestones
Received: 30 March 2018
Revised: 10 September 2018
Accepted: 28 October 2018
Published: 16 April 2019

Communicated by Glenn Hurlbert
Authors
Steve Butler
Department of Mathematics
Iowa State University
Ames, IA
United States
Misa Hamanaka
Department of Mathematics
Iowa State University
Ames, IA
United States
Marie Hardt
Department of Mathematics
Iowa State University
Ames, IA
United States