Research of NP-Complete Problems in the Class of Prefractal Graphs

Round 1

Reviewer 1 Report

Review report on the article: " Research of NP-complete Problems in the Class of Prefractal Graphs. "
The paper proposes a study of some well-known NP-complete problems on a special class of graphs - prefractal graphs. The problem of NP-completeness in graph theory is well known and sufficiently studied. Nevertheless, the problem remains open, and researchers propose different methods and consider special cases of problem-solving.
The introduction provides an overview of publications on the NP-completeness problem and provides the main definitions of prefractal graphs with the author's references to earlier publications. At the same time, it would be interesting to know the difference between prefractal graphs and fractal ones; it would be desirable to add information about this.
In the second section, the author study particular statements of NP-complete problems on pre-fractal graphs. That is, under what conditions it is possible to give an answer about the existence of a solution. Algorithms for finding such solutions have been proposed for some problems. In general, this approach corresponds to the canons of studying NP-complete problems on graphs.
For example, in the problem of isomorphism to a subgraph, the conditions for constructing two prefractal graphs, under which they will be isomorphic, are distinguished.
An interesting result is the use of existing algorithms as procedures in the new author's algorithms. Thus, it is possible to apply emerging new algorithms to improve the characteristics of the author's algorithms. This feature became possible due to the properties of prefractal graphs.
The second section examines six problems: Subgraph isomorphism; Degree constrained spanning tree; Maximum leaf spanning tree; Balanced complete bipartite subgraph; Bipartite subgraph; Planar subgraph. For each problem, some results are obtained.
The third section presents the main results of the study, including the computational characteristics of the developed algorithms. The execution time of the proposed algorithms is polynomial and depends linearly on the number of vertices in the graph. In this case, the estimate contains a parameter - a constant for the algorithm, which is defined as the number of seed vertices. It is correct to call such an algorithm - a polynomial algorithm with a parameter or a parameterized polynomial algorithm. In this sense, corrections should be made to the text of the article.
The Conclusions section (numbering is broken; you need to put the number 4) provides a small overview of the prefractal graphs' areas. And also what benefits can be obtained using the prefractal graph toolkit.
Regarding this article, the following issues can be noted:
1) in the introduction, add a mention of self-similar graphs (several links to publications) and how they are related to prefractal graphs;
2) in the results to generalize the question of what the study of all known NP-complete problems on prefractal graphs will lead to; what is the methodological result;
3) add some links to articles on dynamic graphs in the introduction; add information in the conclusion about the planned studies of dynamic graphs and how they relate to prefractal or fractal graphs;
4) fractal graph - essentially an infinite graph, how the proposed algorithms will work on fractal graphs; add information about this at the Conclusions of the article.
In general, the work corresponds to the topics of the journal Mathematics. The proposed results are new and interesting. According to preliminary searches, the materials of the article are for the first time offered for publication and will be useful to researchers interested in this issue. After the elimination of the remarks indicated in this review, the article can be recommended for publication.
It is also recommended that the author publish a separate article on fractal and prefractal graphs in the future in order to familiarize the scientific community with this class of special graphs and consecrate their properties and characteristics.

Author Response

Dear Reviewer!
Thank you for your detailed comments and recommendations. "Response to Reviewer 1 Comments" are attached as a file (Word).

With Best regards, Rasul Kochkarov

Author Response File: Author Response.docx

Reviewer 2 Report

The paper studies six known NP-complete problems in relation to the class of prefractal graphs: isomorphism of a subgraph, degree constrained spanning tree, maximum leaf spanning tree, balanced complete bipartite subgraph, selecting a bipartite subgraph, planarity of a prefractal graph. For each of the problems, solvability conditions are given under which it is possible to obtain the answer of solution existence for some subproblems. Polynomial algorithms to construct solutions are also given. Some of the problems are simple and their proposed methods are intuitive, e.g., the proposed algorithm for finding a spanning tree. Nevertheless, the paper is consistent, interesting, well-written, and easy to follow. Therefore, I propose the paper to be accepted for publication in Mathematics.

Author Response

Dear Reviewer!
Thank you for your Comments and Suggestions. The changes made to the article are reflected in the file (Word).

With Best regards, Rasul Kochkarov

Author Response File: Author Response.docx

Reviewer 3 Report

Dear Authors.

This paper (Research of NP-complete Problems in the Class of Prefractal Graphs) is interesting work, emphasizing how important for the NP-complete Problems. It is good paper. However, I have some Major comments.

Strengths:
1.  An interesting partition of NP-complete Problems.

2. This paper is interest to the readers.

Weaknesses:

1.
Reason behind deploying in a "NP-complete Problems" should be elaborated more.

2. References are good. But, new references need...
Important references are missing:

2.1) "Some simplified NP-complete problems." Proceedings of the sixth annual ACM symposium on Theory of computing. 1974.

2.2) Some NP-complete problems in quadratic and nonlinear programming. 1985.

2.3) Some NP-complete problems on graphs. No. CS Technion report CS-2011-05. Computer Science Department, Technion, 2011.

2.4) "Finding Innovative and Efficient Solutions to NP-Hard and NP-Complete Problems in Graph Theory." Computer Science 5.2 (2020): 137-143.

2.5) "Hierarchical System Decomposition Using Genetic Algorithm for Future Sustainable Computing." Sustainability 12.6 (2020): 2177.

2.6)  "New iteration parallel-based method for solving graph NP-complete problems with reconfigurable computer systems." IOP Conference Series: Materials Science and Engineering. Vol. 919. No. 5. IOP Publishing, 2020.

3.
3.1. What is NP-complete Problems ? (Study more)
3.2. What is Class of Prefractal Graphs ? (Study more)
3.3. What is main idea ?
3.4. What is Improved ?

4. 
Control missing in some experiments.
-More experiments.

5. English: Very good, minor spell check.
English language and style are fine/minor spell check required

6. Scientific Soundness
Scientific Soundness: Average

Author Response

Dear Reviewer!
Thank you for your detailed comments and recommendations. "Response to Reviewer 3 Comments" are attached as a file (Word).

With Best regards, Rasul Kochkarov

Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

Dear Authors

 

NP-complete Problems in the Class of Prefractal Graphs lacks logical validity as a good topic or paper. However, not all of the reviewer's requests were taken into account.

 

Also, paper has serious flaws, additional experiments needed, research not conducted correctly.

Author Response

Dear Reviewer,
Thanks for your comments and remarks. The answer is attached as a pdf file.

Best regards, Rasul Kochkarov

Author Response File: Author Response.pdf