Biequivalent Planar Graphs

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

See attachment

Comments for author File: Comments.pdf

Comments on the Quality of English Language

English Language is fine. There are some typos in the text.

Author Response

We would like to thank the Referee for their remarks that have helped us make
our manuscript significantly better. In what follows, we shall outline the changes
we have made for each point raised. We hope that with these implemented
changes the referee will find our manuscript suitable for publication in Symmetry.
• 1. It would be desirable to add a review of similar theoretical works on
graph structures to the review of articles on protein cages.
The study of the structure of protein cages is quite new and most investigations
have so far been mostly experimental. The formal mathematical
description of polyhedral cages as well as the connection between their
structure and planar graphs were identified in our Proceeding of the Royal
Society paper (Piette2022). We are not aware of other similar work related
to the mathematical description of protein cage structure. We have
added a paragraph in the introduction to describe this.
• 2. In continuation to the previous comment. The text has a very advantageous
illustration part, but a rather simple mathematical apparatus.
Perhaps the mathematical part should be strengthened, including focusing
on similar graph-theoretic studies.
Our aim was to identify all graphs with the desired symmetry and the
method we have used is by construction together with some algebraic considerations.
There might be other methods to achieve the same goal, but
this is the one we have used.
• 3. P. 2, Formulae (3)-(4). In these formulas it is necessary to mention
that the outside face is taken into account. The first mention of this fact
appears only on page 5. Before that, the reader gets the impression that
the formulas are wrong.
This is a very good point and we have added a sentence page 2 to emphasie
that point.
• 4. The rest of the comments are more about technical details and typos.
They are highlighted.
We thanks the referee for spotting all these minor mistakes ans we have
corrected them all.
Bernard Piette
1

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This is an interesting article dealing with bi-equivalent planar graphs.  Author used an algebraic method followed by an algorithmic procedure to determine all such graphs.  The graphic representation of every graph found was presented.  Is this appropriate way to demonstrate all these tiling patterns in this Journal?

Author Response

I would like to thank the Referee for their positive comments on the paper and
we answer the question raised in their report as follows:
• The graphic representation of every graph found was presented. Is this
appropriate way to demonstrate all these tiling patterns in this Journal?
A graph is really described by the connections between the different nodes,
or more explicitly, by a set of pair of nodes which are linked together.
Printing such list in a journal would be very tedious and somewhat not
very informative. As described in the paper, the files containing these pair
of links for each presented graph is available from the material we have
archived in Zenodo.
The most intuitive methods to present a graph is using a graph and this is
the method we have chosen. Moreover to construct polyhedral cages, the
main motivation for our study, such graphic representations are necessary.
As a matter of fact the graphical representation of the graph depends on the
chosen outside face, but as there were already many graphs in the paper, to
present each graph for all choice of outside face would have made the paper
far too long. We have thus chosen for the outside face the one leading to
the clearest representation (in most case the largest polygon).
I have added a paragraph in section 4 to emphasise the point made above.
Bernard Piette
1

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The author defines bi-equivalent planar graphs as generalization of the uniform polyhedron graphs with 2 families equivalent nodes. In the present article the author gives a list of bi-equivalent planar graphs with less than 300 nodes with valency between 3 and 6 by using computer program. The author explains that the research in this direction has applications to artificial protein cages which are developed for targeted drug delivery. The result is clearly presented.

Author Response

I would like to thank the Referee for their positive comments on the paper.
Bernard Piette

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Still I have question about presenting all the graphs in the paper.  Most of the graphs should be collected in supplementary materials.  Only important classes of graphs are presented.  Besides, the algorithm /program should be added or described.  Revisions are still required.

Author Response

I would like to thank the Referee for their positive comments on the paper and
we answer the question raised in their report as follows:

• Still I have question about presenting all the graphs in the paper. Most
of the graphs should be collected in supplementary materials. Only important
classes of graphs are presented.

I have now removed from the paper all the graphs but a few representatives
of the different families of graphs. This has halved the length of the
paper. A complete list of all the graphs is available in the supplementary
material.

• Besides, the algorithm /program should be added or described. Revisions
are still required.

The full program used to compute these graphs is available from Zenodo
and the link is given at the end of the paper. I have added a section to the
paper to better describe the algorithm used to construct the graphs.

Bernard Piette

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

I just went through the newly revised version.  I seems fine to me that it should be accepted now.