Edge Neighbor Toughness of Graphs
Round 1
Reviewer 1 Report
The authors introduce a new toughness parameter, which they first compute for classical graph topologies. The main result in this respect is obtained for complete multipartite graphs. This is an elegant result. However, the authors incorrectly replaced their notation n_p by X_p (in the proof and the subsequent remarks/examples), that must be corrected.
An NP-hardness proof for computing this parameter is sketched, but the main technical argument is deferred to a proof in another paper. This part of the proof needs to be completed before the paper can be accepted.
Finally, the authors provide a simple polynomial algorithm for trees. The runtime is O(n^2) being given the adjacency matrix, which is optimal. However, one usually considers adjacency list representation, and then the runtime can be lowered to O(n). This should be mentioned.
The proofs are not terribly difficult, but I think that it can be a decent contribution to this journal (provided the minor comments which I listed above are taken into account).
Observation: in general, the \omega notation stands for the clique-number, bot for the number of connected components. It would be better coming with another notation.
Author Response
Dear professor, we have revised our manuscript and submited the new version to the editorial office. The detail is in the cover latter attached below.
Thank you very much!
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The manuscript “Edge neighbor-toughness of graphs” is devoted to the study of the invulnerability of communication and other networks by measuring the special characteristics of the underlying graphs.
The authors define the parameter called edge neighbor-toughness of a graph and discuss the properties of this parameter. Theorems on the value of this parameter for graphs of some standard topologies are formulated and proved: path, cycle, complete graph, complete p-partite graph, comet.
It is proved that the computational problem of edge neighbor-toughness of a graph is NP-complete. The authors proposed a polynomial algorithm for computing the edge neighbor-toughness of trees.
The article is written in a good competent scientific language.
Comments
Most of the comments are just proofreading notes, but some of them need attention.
- P. 1. Abstract: “to measures…” – “to measure…”
- P. 1. Abstract: “few edges.” – “a few edges.”
- P. 1. Introduction: “network as a whole.” – “the network as a whole.”
- P. 2. “The edge-neighbor-scattering number of G is defined as [4] ENS(G) = max{\omega(G/S) –|S|}…” and further in this paragraph. Throughout the article, both above and below, the edge cut-strategy is denoted as X. It is undesirable to deviate from a uniform notation.
- P. 4. Theorem 4. The notation "X_p" is not explicitly defined. This must be done in or before the statement of the theorem.
- Section 4 is better divided into two parts: the analysis of the parameters and the сonclusion itself. The conclusion should be extended by reviewing the results obtained and outlining the field of further research.
- And the last comment. Typically, annotations do not contain formulas. Annotations, like keywords, belong to the metadata of articles and should be available for automatic processing.
Wish you luck
Author Response
Dear professor, we have revised our manuscript and submited the new version to the editorial office. The detail is in the cover letter attached below.
Thank you very much!
Author Response File: Author Response.pdf
