Graphs with at most two trees in a forest building process

TL;DR

This paper analyzes a process for forming spanning forests in graphs based on edge orderings, focusing on probabilities of resulting in one or two trees, and constructs infinite families of non-isomorphic graphs with identical probabilities.

Contribution

It characterizes the probabilities of forming at most two trees in the forest building process and constructs infinite non-isomorphic graph families with identical probabilities.

Findings

Probability of forming one or two trees in the process is determined.

Infinite families of non-isomorphic graphs with the same probabilities are constructed.

Abstract

Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep 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 non-isomorphic but produce the same probabilities.