Conditions for Implicit-Degree Sum for Spanning Trees with Few Leaves in K_1,4-Free Graphs

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Manuscript ID: mathematics-2701073
Title: Conditions of implicit-degree sum for spanning trees with few leaves in K1,4-free graphs
Authors: Junqing Cai, Cheng-Kuan Lin, Qiang Sun, Panpan Wang

In this work, the authors considered a K_1,4-free connected n-graph and claimed that for k = 2, 3, if the implicit-degree sum of any k+1 independent vertices of G is at least n−k+2, then G has a spanning k-ended tree. Also, they give two examples to show that 4 the low bounds n and n−1 are the best possible, respectively.

Since this work shows a lack of motivation for such an investigation, it is insignificant for a SCI/SCI-E publication.

Comments on the Quality of English Language

It may be improved.

Author Response

The authors are greatly indebted to the referee of this manuscript for your most helpful comments and suggestions which have led to a substantially improved presentation of the paper. 

We have rewritten the abstract and adding the motivation for this investigation. Please see the detail in the paper colored red.

Reviewer 2 Report

Comments and Suggestions for Authors

The article considers the problem of the existence of spanning trees with a few leaves. The topic of the work is original and relevant to the scientific field of research.

The material in the article is presented in a logical and consistent form. If the abstract contains information about "two examples", then it is better to use the notation Example 1, Example 2 in the text of the article. The Discussion section is too short. It is worth expanding the Discussion and adding Conclusions.

Author Response

The authors are greatly indebted to the referee of this manuscript for your most helpful comments and suggestions which have led to a substantially improved presentation of the paper. The details of our responses to these comments, please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

In the present article the authors obtain  a condition for a certain type of graph to have a spanning tree with few leaves. Let G be a K_{1, 4} free connected graph with n vertices. The authors show that for k = 2, 3, if the implicit-degree sum of any k + 1 independent vertices of G is at least n - k + 2, then G has a spanning k-ended tree. They give examples to show that the above lower bounds are best possible. The authors explain basic notions and refer to [1] for those which are not defined in the article. It would be better to define key notions in the article for the convenience of the readers.

Here is an example.

line 27, Define an independent set of G.

line 62, There is a repetition of "otherwise", and it is hard to see the definition.

Here is a remark about the notation.

line 20, line 102: --> | (for a consistency with other parts)

 

Comments on the Quality of English Language

Moderate editing of English language is required.

Here are some examples.

line 5, low bounds --> lower bounds

line 13, a vertices --> a vertex

line 65, classic results --> classical results

line 104, usefull --> useful

 

Author Response

The authors are greatly indebted to the referee of this manuscript for your most helpful comments and suggestions which have led to a substantially improved presentation of the paper. The details of our responses to these comments, please see the attachment.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The authors presented an interesting paper on degree sum conditions for spanning trees with few leaves in K1.4-free graphs. The article will be convenient for publication after taking into account all factual and editorial comments 

1. INTRODUCTION 

Could you discuss the definition of 'implicit-degree' proposed by Zhu, Li, and Deng [19]? How do you understand this definition and what role could it play in graph analysis? Do you see any connection between implicit-degree and other mathematical concepts discussed earlier in the text?"

Could you explain the relationship between the definition of implicit-degree and the conditions for the existence of k-ended trees in graphs free from certain subgraphs (e.g., K1,3-free or K1,4-free) discussed in the article? Is there any dependency or implication between the conditions of the theorems and the concept of implicit-degree?"

2. Preliminaries

What role do Lemes 2.1, 2.2, and 2.4 play in the context of path P in graph G? Can you explain how these lemmas relate to the conditions of existence of k-completed trees in the context of the P path and what conclusions come from these lemmas in graph analysis?

Could you explain how the conditions in Remark 3 for graphs G' and G'' relate to the definitions σ3(G') = 3m = n − 1, iσ3(G') = 2m + (m + 1) = n and σ4(G'') = 4m = n − 2? Do these equalities have practical consequences for the possibility of Hamilton's paths and k-trees ending in these graphs?

I wonder if the theory presented in Remark 3 fully coincides with the actual calculations for the G' and G'' graphs. Are the equals σ3(G') = 3m = n − 1, iσ3(G') = 2m + (m + 1) = n and σ4(G'') = 4m = n − 2 are absolutely certain and fulfilled in practice? Is there a possibility of errors or imprecisities in these calculations that could affect the conclusions about the existence of Hamilton's paths and k-trees terminated in these graphs?

3. Proof of Theorem 

Is it possible to explain why, when choosing the maximum T tree with exactly 4 leaves, we minimize ∆(T)? Does this minimization play a key role in proving Theorem 1.5? Are there other approaches to this problem that could also lead to similar conclusions?

Can we reflect on the certainty and effectiveness of the steps in the proof of theorem 1.5? Is there a possibility of errors or gaps in reasoning leading to conclusions? Are these steps absolutely certain and not subject to doubt, or is there a chance of inaccuracies or errors in argumentation?

Discussion about the sufficient conditions of existence of spreading trees with a small number of leaves is an important point of this article. An extension of this discussion may include an in-depth discussion of the practical applications of tree breakout with a limited number of leaves in various fields, such as communication networks, optimization algorithms, and efficient routing.

REVIEW SUMMARY

Expanding the article to include more examples of practical applications of scanning trees with a limited number of leaves would definitely be a valuable addition. One could consider adding specific case studies from fields where such graph structures are used, such as in communication networks, route planning, or data analysis in the social sciences.

In addition, they could expand the discussion to include:

• Computational Complexity Comparison: A more detailed analysis of computational complexity for different classes of graphs to better understand where these sufficiency conditions are more effective.
• Exploring Boundary Conditions: Examining the limiting cases in which these conditions are not satisfied could provide important insights into the limitations of these conditions.
• Reviewing Methods for Extending Conditions: Comparing these conditions with other known methods for determining the existence of trees with specific properties in graphs could provide a deeper understanding of their effectiveness and universality.
• Examples of Applications in Modern Technologies: Pointing out specific applications in the context of modern technologies, such as neural networks, machine learning algorithms, and Big Data analytics

After the above corrections, the article will be convenient for publication.

The article will be convenient for publication after taking into account all factual and editorial comments 

 

Comments on the Quality of English Language

minor corrections

Author Response

The authors are greatly indebted to the referee of this manuscript for your most helpful comments and suggestions which have led to a substantially improved presentation of the paper. The details of our responses to these comments, please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

Comments and Suggestions for Authors

In my opinion, the current article has been corrected and is suitable for publication 

Comments on the Quality of English Language

perhaps a small correction