Properties and Recognition of Atom Graphs
Garth Isaak
Round 1
Reviewer 1 Report
See attached report.
Comments for author File: 
Comments.pdf
Author Response
We thank Revewer 1 for his commznts.
In the preliminaries, ws have recalled ithe definitions concerning separation, because some of them aere not so standaed (components of S in G, full component…). Moreover, ther may be ambiguity between « minimal seoarator » and «inclusion-minimal seoarator », and about the empty set being a separator of a disconnecred graph or nor (it is in [23], but not in our paper, so we had to add the word « connected » in Characterization 2.4 from [23],)
We have left the definition of a chordal graph because it is very short.
We do not understand the remark about Definition 2.4, which is in fact Characterization 2.4. But as this characterization was written in the form of a definitionn we have reformulated it into a characterization, with « if and only if »..
We have made all other suggested corrections/modifications.
Reviewer 2 Report
In this paper, the authors focus on atom graphs, their properties and their recognition. In particular, an algorithm for the recognition of an atom tree is divulged.
In the opinion of this reviewer, this paper is suitable for publication in this journal subject to a minor revision.
The main difficulty of this reviewer while reading this paper was realizing the difference between 'an atom graph' and 'the atom graph of a graph'. The terminology 'atom graph' for both is a little unfortunate, and, at times, confusing. Is the term 'an atom graph' official? If not, perhaps it would be ideal to change it. Particularly, the sentence in the abstract "An atom graph is a graph which is isomorphic to the atom graph of a graph." is very confusing, and seems circular, until one realizes that 'an atom graph' and 'the atom graph of a graph' are two different concepts.
Moreover, Theorem 7.14, which provides the algorithm mentioned earlier, is of order O(mn−1n5). Even though this is an improvement on other algorithms, it is still quite intractable except for very small graphs. A comment on the significance of this result would be welcome.
Furthermore, the authors should also note the following very minor mistakes:
On page 2: deconposition, maxumal, graohs, graoh. Moreover, there is a double comma.
On page 3: usel.
Page 11: notatin.
Page 13: connecvted. Also, there is a double fullstop.
Page 17: expaded.
Page 25: The top sentence ends in a comma, not a fullstop.
Author Response
We thank Revewer 2 for his commznts.
We agree that the sentence in the abstract "An atom graph is a graph which is isomorphic to the atom graph of a graph." is confusing, and we have changed it into « A graph $G$ is an atom graph if there is a graph whose atom graph is isomorphic to $G$. ». We have kept the name « atom graph » since similarlar double fenitio,ns already exist for clique graphs [1] and clique graphs of chordal graphs [25].
We also agree that our algorithm is intractable. We have added the sentence « Note that this algorithm is still quite intractable except for very small graphs. »
We have corrected all typos.
