## Involve: A Journal of Mathematics

- Involve
- Volume 12, Number 4 (2019), 659-670.

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

Steve Butler, Misa Hamanaka, and Marie Hardt

#### 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.

#### Article information

**Source**

Involve, Volume 12, Number 4 (2019), 659-670.

**Dates**

Received: 30 March 2018

Revised: 10 September 2018

Accepted: 28 October 2018

First available in Project Euclid: 30 May 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1559181658

**Digital Object Identifier**

doi:10.2140/involve.2019.12.659

**Mathematical Reviews number (MathSciNet)**

MR3941604

**Zentralblatt MATH identifier**

07072545

**Subjects**

Primary: 05C05: Trees

**Keywords**

forests edge ordering components probability

#### Citation

Butler, Steve; Hamanaka, Misa; Hardt, Marie. Graphs with at most two trees in a forest-building process. Involve 12 (2019), no. 4, 659--670. doi:10.2140/involve.2019.12.659. https://projecteuclid.org/euclid.involve/1559181658