Finding Hamiltonian and Longest (s,t)-Paths of C-Shaped Supergrid Graphs in Linear Time

Round 1

Reviewer 1 Report

Reviewer’s report:  Finding Hamiltonian and Longest (s, t)-paths of C-shaped

Supergrid Graphs in Linear Time

 The authors outline a linear time algorithm for enumerating Hamiltonian and longest (3,7)-paths of C-shaped graph. Overall I find this manuscript to be sufficiently interesting but only AFTER revisions as suggested below and another round of review as it contains some serious shortcomings.

 

1. Although the introduction provides some motivation by pointing out 3D-printing as a possible application, the manuscript fails to provide sufficient historical background and several other important applications of the Hamilton-path problem. Actually the authors’ reference only ot Bondy-Murthy book is quite misleading. The problem of Hamiltonian and Eulerian path is a long standing unsolved problem in the most general case, see Ref 1 & 2 (below) of Harary who in fact well before Bondy-Murthy outline important motivations and applications. The authors should cite these refs and provide adequate motivating background as to why the C-shaped supergrid graphs are so important.
2. I would like to point out a recent important applications of Hamilton path problem to Biology that appeared in Nature Communications recently [3]. The authors should cite this work and include such vast applications of Hamilton path problem to increase the impact.
3. Yet another problem of importance in molecular and physical sciences is related to Random walks, enumeration of self-returning walks and ring perception algorithms, peripherals of fullerenes, etc., – all of which relate to the problem studied by the authors, please mention this as well and cite ref [4,5].
4. There are far too many self-citations and the manuscript appears to be more self-serving the authors than an important contribution to the field. Thus out of 24 cited references, 12 belong to papers in which Keshavarz-Kohjerdi is an author and 7 papers in which Ruo-Wei Hung is an author. This is clearly NOT acceptable and in any case this practice of high self-citations in a research article must be strongly discouraged. If that many papers of authors have been cited in a research article where is the originality of this work? It is noted that the authors have studied several alphabet supergrid graphs including the L-shaped supergrid- which is very close to the present C-shaped supergrid. Thus the authors must clearly point out where is the originality and novelty in the present work?
5. Ref 6 listed below seems to bear some relation to the present paper of the authors. Hence the authors must cite that as well and point out any similarities.
6. The authors should point out the connection between Lapalcians of graphs and spanning trees (longest path), as shown with applications ( see ref 4 and works provided therein).
7. The authors do not provide any CPU times to prove that their algorithm is linear in time. Thus they should provide a table of CPU times as a function of s and t for C large enough supergrid graphs C(m, n; k, l; c, d) with mn ⩾ 2, and two distinct vertices s and t in C(m, n; k, l; c, d).
8. English needs improved throughout the manuscript, and some result stated as Theorems hardly deserve the status of theorems, for example, see below:

Theorem 6. Assume that C(m, n; k, l; c, d) is a C-shaped supergrid with two distinct vertices s and t. Then, a longest (s, t)-path in it can be computed in O(mn)-linear time. 730

The linear-time algorithm is given in Algorithm 4.1.

 

Overall: The authors have studied several alphabet supergrid graphs before including L-shaped, A-shaped, and several others. Therefore there is a question as to what the originality of the research presented herein is and how it is different from the authors’ previous recently published works. Is this am incremental addition to the list of alphabets of supergrids or is there any novelty on the present work. This distinction is must be made by the authors and they must provide a very clear and compelling case for the rationale behind studying C-shaped supergrid graphs.

The manuscript should undergo major revision along the lines suggested above and must be evaluated again.

[1] F. Harary. Graph Theory (Addison-Wesley, Reading, MA, 1967).

[2] Hamiltonian Paths in Hypercubes: Harary, F., Hayes, J.P. and Wu, H.J., 1988. A survey of the theory of hypercube graphs. Computers & Mathematics with Applications, 15(4), pp.277-289.

 [3] Twarock, R., Leonov, G. and Stockley, P.G., 2018. Hamiltonian path analysis of viral genomes. Nature communications, 9(1), pp.1-3.

[4] Balasubramanian, K., 2020. Enumeration of stereo, position and chiral isomers of polysubstituted giant fullerenes: Applications to C180 and C240. Fullerenes, Nanotubes and Carbon Nanostructures, 28(9), pp.687-696.

[5] Balasubramanian, K., 1985. Computer-assisted enumeration of walks and self-returning walks on chemical graphs. Computers & chemistry, 9(1), pp.43-52.

[6]Chen, X.E., 2020. Vertex-Distinguishing E-Total Coloring of Complete Bipartite Graph K 7, n when n≥ 978. IAENG International Journal of Applied Mathematics, 50(1), pp.1-4.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The problem of finding a linear-time algorithm to compute
a longest simple path between two distinct vertices in these
graphs seems to be interesting in the application to computer
science and in particular to the analysis of 3D printers. Finding
a fast (linear complexity) algorithm to determine a longest simple
path between every two distinct vertices in my opinion, this is a
significant result. Therefore, the article makes a very favorable
impression.   However, there are questions that the authors would like to raise.
Firstly, a very large volume of the article - 23 pages. Is it
possible to reduce it to at least 15 pages, while preserving the
most important result. Secondly, is it possible by the authors' algorithm to find all
simple paths of maximum length between a pair of vertices.
Thirdly, modifications of Floyd's algorithm are known to calculate
the maximum length of simple paths between all pairs of digraph
vertices. Is it possible to construct a modification of Floyd's
algorithm to calculate the matrix of maximum lengths of simple
paths between all pairs of vertices in the graph considered by
the authors so that in some sense such an algorithm would be fast
enough and evaluate its computational complexity?

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

The authors build on the work of Hamiltonian graphs. I ntheir paper the authors first study the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of rectangular supergrid graphs with a hole. Next, they show that C-shaped supergrid graphs are Hamiltonian connected except few conditions. Finally, they propose a linear-time algorithm to compute a longest path between two distinct vertices in these graphs.

 

Overall I found the article very interesting and the authors have done an excellent job in writing this manuscript. I would suggest the following minor comments that you might want to take into consideration. I feel that in the conclusion section you should incorporate some discussion o the implications of your work for other fields.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have thoroughly revised the manuscript. Although I would have liked to see the actual CPU times of the authors' algorithms, I will take their word that it is linear in time. Hence I am recommending publication as is.