Feasible Edge Replacements - Class and Algorithms

About

This repo contains four things: A class for Fer objects in feasible_edge_replacement.py. Some general methods for the study of amoeba graphs in amoebas.py. In particular, a method for finding Fer generators (called updating_Cayley_populate), something that is referenced but left out of [1]. An implementation of the algorithm of [1] in main.py. A construction of a recursive family of amoebas to apply the algorithm to, for illustrative purposes, in treebonacci.py.

Every Fer object has two attributes: a sequence of feasible edge replacements and its corresponding permutation. The length of a sequence can be obtained by len(fer). The product * of two Fer objects multiplies their permutations and concatenates the edge replacements, updating the labels.

Note that products are written left-to-right, in contrast with traditional algebraic notation.

The algorithm produces a hash map that links every permutation in a generating set of the symmetric group to a Fer object.

Mathematical background

Let $G$ be a graph and $e$ an edge of $G$. We say $(e\to g)$ is a feasible edge replacement (Fer, for short) if $G-e+g$ is isomorphic to $G$. We say a graph is an amoeba is it can be transformed into any isomorphic copy of itself by means of a sequence of Fer objects. For more details on amoebas and the difference between local and global amoebas, consult [2].

This repo offers a class for Fer objects to study amoebas. Relevant operations are defined for these objects. This class is extensively used in the algorithms presented below.

In the paper [1], it is proved that a certain recursive family of graphs are all amoebas. In this repo, we give an implementation of the algorithm decribed in Section 5 that will factor any permutation of a stem-symmetric (see Definition 6 of [1]) amoeba into Fer objects in time $\Theta(n^2)$, where $n$ is the order of the graph. This succesfully provides an efficient method of finding, for any isomorphic copy of the amoeba, a sequence of replacements that will move the graph into its copy.

Furthermore, this repo contains a construction of one such type of recursive family, called Fibonacci-type trees in [1], to serve as an illustrative example of the algorithm.

Instructions

To use the algorithm, we need to provide a stem-symmetric recursive construction similar to the one given in Theorem 8 of [1]. One such example is provided in treebonacci.py.

Here is a minimal working example on how to use the main features of this repo:

1. Download the folder and leave all Python scripts in the same working directory.
2. Install all dependencies.
3. Open main.py in your favorite editor and provide an input for k and permutation. Default is $k=6$ and random.
4. Output will be the sought sequence.
5. To verify the Fer objects, use fer_verifier in amoebas.py.
6. To change the family of graphs, change line 8 import treebonacci as trb and provide a valid stem-symmetric recursive amoeba family. Note you may need to use the methods in amoebas.py to regenerate the basis objects.
7. To animate the edge-replacements, wait for me to upload the animation module.

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This program was developed by

Tonatiuh Matos Wiederhold Dept. of Mathematics, University of Toronto, Canada. tonamatos@gmail.com Based on the research of [1], below.

References

[1] Eslava, L., Hansberg, A., _, Ventura, D., New recursive constructions of amoebas and their balancing number, preprint, 2023.

[2] Caro, Y., Hansberg, A., Montejano, A. (2023). Graphs isomorphisms under edge-replacements and the family of amoebas. Electronic Journal of Combinatorics 30(3) P3.9.