Subdomination in Graphs with Upper-Bounded Vertex Degree

Round 1

Reviewer 1 Report

This paper deals with the lower bound for the k-subdomination number on the set of graphs with a given upper bound for vertex degrees; there are proposed the optimal graphs and indicate the corresponding k-subdominating functions.

This topic is interesting and actual. It was debated in many papers and the attraction of this approach consists mainly in the manner the authors treated the results
of their quest, respectively results are interpreted in terms of social structures.

The authors add to the subject area an estimation of the number of happy vertices with the quickly approaches of the upper bound  to the minimum possible value, and the social properties of the corresponding graph quickly converge to those of the optimal graph without any limit on the maximum vertex degree. These social characteristics are completely different from those of regular graphs.

Authors should add to the paper computer simulations of their social model. Such approach will allow discussing possible real values for the upper-bound related to the graphic size and the vote quota.  

Conclusions are not relevant as authors wrote “…each member of the society must maintain a sufficiently large number of social
contacts (connections in the social graph).”

Question is what exactly means as much as possible?  Computer simulations with possible numeric values of the involved parameters should answer this question.

Although the references are appropriate new articles related in literature should be added, as stated below, see point 7.

I also recommend the authors to follow the next suggestions:

1.     Authors should add short article content at the end of the chapter Introduction.

2.     State of the art should be upgraded for revealing the technical novelty and originality of the proposed approach as well as the advantages vs disadvantages of it.

3.     All equations should be justified by references citations or by mathematical proofs.

4.     All equations should be labeled.

5.     Authors should explain the relations (5) and (6).

6.     Authors should mention in chapter Conclusion further development of their approach.

7.     Authors should add “fresh” references from the last five years.

8.     An English native speaker should correct the grammar and spelling.

An English native speaker should correct the grammar and spelling.

 

Author Response

Dear reviewer,

Thank you very much for careful reading and benevolent treatment of our manuscript. Please find below the answers to your comments.

To meet your comment on computer simulations of our social model to discuss possible real values for the upper-bound related to the graphic size and the vote quota we added the computer simulation of the proposed lower bound for different graph sizes, upper bound of graph degree, and the vote quota.   

To address your comment on the conclusions we reformulated the conclusion to exclude the words "as much as possible". 

References to new articles related in literature are also added.

Also the following changes were made:

1. The short article content was added at the end of Introduction section.
2. Technical novelty and originality of the proposed approach were highlighted in the end of the Review section, while the discussion of advantages vs disadvantages is added to the Conclusion. 
3. All equations are justified by references or by mathematical proofs.
4. According to Mathematics journal material preparation policies, all equations which referenced in the text are labeled.
5. Relations (5) and (6) are explained in the text.
6. Further development of the approach and promising lines of research are discussed in the Conclusion.
7. References from the last five years are added to the article.
8. An English native speaker checked grammar and spelling.

Reviewer 2 Report

Comments.

1 An interpretation of the investigation in terms of social structures is appropriate. But there is no connection with US voting. I recommend one more paper

Semenov A., Gorbatenko D., Kochemazov S. Computational study of activation dynamics on networks of arbitrary structure. Computational aspects and applications in large-scale networks, 205–220, Springer Proc. Math. Stat., 247, 2018. 

2. Misprint. Extra e in line 70

3. |N(v)|=deg v+1 in line 49 but |N(v)|=deg v in line 70.

4. Is \Delta the maximal vertex degree or degree +1?  Are there a loop in every vertex?

 

 

Author Response

Dear reviewer,

Thank you very much for careful reading and benevolent treatment of our manuscript. Please find below the answers to your comments.

1. To avoid misunderstanding reference to US voting was removed from the text.
2. References to papers on conformal agents' voting are added.
3. Misprints are corrected.
4. As noted in the last paragraph of Section 2, "We consider only graphs
   with each vertex having a loop", so the vertex degree in our article includes this loop and |N(v)|=deg v. This notation is common in graph domination literature, and we could not find any confusion of different degree notations in our text.

Reviewer 3 Report

Excellent introduction to the theme of the invariant subdomination in graphs with original results on the non-simple relation between lower bound for k-subdomination number & vertex degrees upper bound.

The reported applications to "political science" may be of some general interest. A possible open point for future investigation is the persistence of some structure by varying the scale of the interaction in large social graphs.

 According with this referee the article is good for publication in the present form.

 

okay

Author Response

Dear reviewer,

Thank you very much for careful reading and benevolent treatment of our manuscript.

The authors