Center of Trapezoid Graph: Application in Selecting Center Location to Set up a Private Hospital

For Reviewer 1

Reviewer 1’s comments/suggestions

Reply to honourable reviewers

The paper is interesting and well presented, and the research is a good fit for the journal.

Thanks for this comment.

The standard of English is good, although there are some minor wobbles.

We have thoroughly checked our revised paper consulting with an expert in English. We also checked our paper through https://www.grammarly.com. All changes are highlighted by red colour.

In particular in the references, please make sure that your style is consistent, e.g. in use of journal abbreviations and capital letters in paper titles.

We have rewritten the References section of our revised paper as per the MDPI journal’s template.

Also some of the terminology seems to me to be non-standard, e.g. rad, diam and ecc (or just e) are more typical

We changed these symbols in our revised paper as following.

We replaced the symbols ‘dmtr’ by ‘diameter’; ‘diam’ by ‘diameter’;  ‘ecnt’ by ‘eccentricity’ ; ‘ecc’ by ‘eccentricity’; ‘rads’ by ‘radius’; We also used some new notations like ‘L(u)’, ‘pnode’, ‘a’ ,  ‘b’, ‘d_{x,i} ’ and ‘e_i’ in our revised paper.

Most importantly though, there is a reference that needs to be taken into account, namely 'Radius versus diameter in cocomparability and intersection graphs' by Erich Prisner (1997). This article proves that 2*rad-diam \leq 2 for a trapezoid graph, which is actually slightly stronger than one of the main theorems in the present paper.

We have included this reference (paper by Erich Prisner) in our revised paper (reference number 38). Erich Prisner established an important result  for trapezoid graph . This result was slightly stronger than our result (Corollary 2 in our submitted paper). In our revised paper, we have improved our previous result and established the Prisner’s result (see Corollary 5,  in revised paper) with the help of four corollaries,  Corollary 1, 2, 3 and 4 (presented in revised paper).  In Corollary 1 and 3, our result () is slightly better than Prisner’s result ().

For Reviewer 2

Reviewer 2’s comments/suggestions

Reply to honourable reviewers

1.      The abstract should be rewritten to encompass all the innovations and contributions of the paper.        

We have rewritten the abstract of our paper mentioning all the innovations and contributions of our paper. All changes are highlighted by blue colour.

2.      The original contributions should be presented more clearly in the final paragraph of the INTRODUCTION section. Any improvements or new findings should be thoroughly described in this part, as the text does not currently address the advantages of the work.

We have presented our paper’s main results and contributions in the final paragraph of the INTRODUCTION section. It is highlighted by blue colour.

3.      A brief outline of the paper’s structure could be added at the end of the Introduction

We have edited the Subsection 1.4 where we present a brief outline of our paper’s structure.

4.      The novelty of the numerical method in this paper is quite limited. What is the main novelty of this method in this article compared to other articles?

1.     In our paper, we have found the radius, diameter and center of trapezoid graph. It is well known that to determine the graph’s center and diameter it is essential to compute the eccentricity of each node. But, we have showed (in Corollary 6) that to find the center of a trapezoid graph it is not necessary to find the eccentricity of all vertices.

2.     We also established a relationship  (in Corollary 5) between radius(G) and $diameter(G)$ of trapezoid graphs and show that in two cases () our results are better than the result of Prisner et al. [38].

3.     To the best our knowledge, to find the center of general weighted graphs we need  time. We have find radius, diameter and center of trapezoid graphs in  time with the help of BFS graphs traversal technique.

4.      Most of the researchers [2, 8, 23, 26, 31,32,36, 38] computed the radius, diameter and center of graphs like interval graphs, permutation graphs, sparse graphs, chordal graphs, etc., but they did not provide any real application in their papers. We have solved algorithmically a real facility location problem with the help of trapezoid graph model using BFS graph traversal technique in  time. To the best of our knowledge, the healthcare facility location problem and other types of facility location problems have not been solved by the Breadth-first search method.

These are the main novelty of our paper.

5.      The references list is not updated with the latest developments and publications. Therefore, I suggest the authors to keep up to date with the relevant literature: on the application: https://doi.org/10.22436/jmcs.029.03.04

We have improved the literature review section of our paper. We have updated the references by including 15 new references [6, 14, 15, 21,27, 35, 38, 40, 41-49]. We did not include the reference https://doi.org/10.22436/jmcs.029.03.04 as we did not find any connection with our paper.

6.      The authors should specify which software was used to produce the results presented in the paper

We did not use any software to get our results. We have proved all Lemmas and theory mathematically. We have calculated the time complexity of our proposed algorithm mathematically with the help of data structure of trapezoid graphs.

(7.1) ”These problems often arise in transportation, urban planning, and distribution network design.” should be ”These problems often arise in transportation, urban planning, and distribution network design.”

Reviewer’s suggestion is same with our previous writing.

(7.2 ”Suppose T be a set of n trapezoids t1, t2, t3, . . ., tn.” should be ”Suppose T is a set of n trapezoids t1, t2, t3, ..., tn.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.3) ”In trapezoid diagram (T-diagram), we define a trapezoid ti by four different corner points [pi, qi, ri, si ].” should be ”In a trapezoid diagram (T-diagram), we define a trapezoid ti by four different corner points [pi, qi, ri, si].”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.4) ”Usually a T-diagram [12] has two parallel horizontal lines – one is top line and other is bottom line.” should be ”A typical T-diagram [12] has two parallel horizontal lines – one as the top line and the other as the bottom line.”

 We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.5) ”we assume that no two trapezoids have a common vertex.” should be ”We assume that no two trapezoids share a common vertex.” 

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.6) ”In T-diagram, we place the corner points pi and qi (ff pi ) on the top line and ri and si (ff ri ) on the bottom line.” should be ”In the T-diagram, we place the corner points pi and qi (ff pi) on the top line, and ri and si (ff ri) on the bottom line.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.7) ”The trapezoids in the T-diagram are indexed based on the ascending order of their corner points qi, i = 1, 2, ..., n.” should be ”The trapezoids in the T-diagram are indexed based on the ascending order of their corner points qi, where i = 1, 2, ..., n.” 

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.8) ”A trapezoid graph is the intersection graph formed by a set of trapezoids arranged within a T-diagram.” should be ”A trapezoid graph is an intersection graph formed by a set of trapezoids arranged within a T-diagram.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.9) ”Here we take an undirected trapezoid graph which is also simple and connected.” should be ”Here, we consider an undirected trapezoid graph, which is also simple and connected.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.10) ”Two trapezoids ti and tj, (j > i) intersect each other iff either (pj − qi ) < 0 or (rj − si ) < 0 or both.” should be ”Two trapezoids ti and tj (j > i) intersect if and only if either (pj − qi) < 0, or (rj − si) < 0, or both.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.11) ”The recognition of a trapezoid graph can be accomplished in O(n2) time [22].” should be ”The recognition of a trapezoid graph can be accomplished in O(n²) time [22].”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.12) ”Initial studies on trapezoid graphs were presented in [11,12].” should be ”Initial studies on trapezoid graphs were presented in [11, 12].”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.13) ”This graph is more complex than permutation as well as interval graph [17].” should be ”This graph is more complex than both permutation and interval graphs [17].”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.14) ”It is also subclass of co-comparability graphs [21].” should be ”It is also a subclass of co-comparability graphs [21].”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.15) ”We draw a trapezoid diagram and its corresponding trapezoid graph in Figure 1 and Figure 2 separately.” should be ”We present a trapezoid diagram and its corresponding trapezoid graph in Figures 1 and 2.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.16) ”For convenience, we use the symbol ta and tb to indicate the trapezoid whose bottom left corner point position is minimum and the trapezoid whose bottom right  corner point position is maximum respectively.” should be ”For convenience, we use the symbols ta and tb to indicate the trapezoid whose bottom-left corner point position is minimum and the trapezoid whose bottom-right corner point position is maximum, respectively.”  

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.17) ”This graph has many applications like Emergency Services: Finding optimal location for fire stations or hospitals to minimize the response time to the farthest service area;” should be ”This graph has many applications, such as Emergency Services: Finding the optimal location for fire stations or hospitals to minimize response time to the farthest service area;”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.18) ”Telecommunication Networks: Placing cell towers or communication hubs to maximize coverage while minimizing the maximum distance to undeserved regions and Warehousing and Distribution: Choosing warehouse locations to minimize the delivery distance to the furthest retail outlets etc.” should be ”Telecommunication Networks: Placing cell towers or communication hubs to maximize coverage while minimizing the maximum distance to underserved regions; and Warehousing and Distribution: Choosing warehouse locations to minimize the delivery distance to the furthest retail outlets, etc.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.19) ”the eccentricity of a vertex v ff V, we denote it by ecnt(v) represents the greatest length among all shortest paths starts from v and finish at other remaining nodes” should be ”The eccentricity of a vertex v ∈ V , denoted as ecnt(v), represents the greatest length among all shortest paths starting from v and finishing at other remaining nodes.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

We have also changed the symbol ‘ecnt(v)’ by ‘eccentricity(v)’.

(7.20) ”ecnt(v) = supd(v, u1) : u1 ff V” should be ”ecnt(v) = sup{d(v, u1) : u1 ∈ V }”  

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.21) ”The maximum among all eccentricities of the nodes of G is considered as the diameter of G” should be ”The maximum among all eccentricities of the nodes of G is considered the diameter of G.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.22) ”Also, the least eccentricity of all node points of G is assumed as the radius of G” should be ”Also, the least eccentricity of all nodes of G is assumed to be the radius of G.” 

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.23) ”A central vertex/node z belongs to G is a node such that ecnt(z) = rads(G)” should be 4 ”A central vertex/node z belonging to G is a node such that ecnt(z) = rads(G).”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

We have also changed the symbol ‘rads(G)’ by ‘radius(G)’.

(7.24) ”A graph may contain one or more central vertices” should be ”A graph may contain one or more central vertices.”

Reviewer’s suggestion is same with our previous writing.

(7.25) ”The set of all central nodes of a graph G we call as the center of G” should be ”The set of all central nodes of a graph G is called the center of G.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.26) ”Computation of diameter, radius and center of a graph is fundamental and it has so many real life applications” should be ”Computation of the diameter, radius, and center of a graph is fundamental and has many real-life applications.” 

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.27) ”In different fields such as facility location problem [33], social network problem [1], biological system [1], transportation network [1] etc., we can apply the concept of the center of the graph” should be ”In different fields such as the facility location problem [33], social network problem [1], biological systems [1], and transportation networks [1], we can apply the concept of the center of the graph.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.28) ”Behzad et al. [6] first established a relation between radius and diameter of connected graph” should be ”Behzad et al. [6] first established a relation between the radius and diameter of a connected graph.”   

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.29) ”Saha [35] also presented parallel algorithms (takes O((n2/p) + logn)-time on a EREW PRAM, where p represents the number of processors)” should be ”Saha [35] also presented parallel algorithms (taking O((n 2/p) + log n)-time on an EREW PRAM, where p represents the number of processors).”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.30) ”Michele et al. [7] have solved the center problem on weakly connected digraph within O(|V||E|) time” should be ”Michele et al. [7] solved the center problem on weakly connected digraphs within O(|V ||E|) time.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.31) ”In [1], an efficient algorithm was proposed to identify the center of a graph, running in O(n3−a) time, where a > 0.01 represents constant” should be ”In [1], an efficient algorithm was proposed to identify the center of a graph, running in O(n 3−a ) time, where a > 0.01 is a constant.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.32) ”Also, Abound et al. [2] introduced another faster algorithm that detects the center of a 5 sparse graph in O(n2−ff) time via solving the all pairs shortest paths” should be ”Also, Abound et al. [2] introduced a faster algorithm that detects the center of a sparse graph in O(n 2−δ ) time via solving all pairs shortest paths.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.33) ”Laskar et al. [20] established a relationship between the rads(G) and dmtr(G) of the chordal graph G” should be ”Laskar et al. [20] established a relationship between rads(G) and dmtr(G) of the chordal graph G.”

Reviewer’s suggestion is same with our previous writing. We have only changed the sumbols ‘rads’ and ‘dmtr(G)’.

(7.34) ”In 2021, Nandi et al. [26] created an O(n) time algorithm for calculating the rads(G), dmtr(G) and center of permutation graphs” should be ”In 2021, Nandi et al. [26] created an O(n) time algorithm for calculating rads(G), dmtr(G), and the center of permutation graphs.”

We have written as per the reviewer’s suggestion and highlighted it by blue colour.

(7.35) ”We have also established a new sharp relation between rads(G) and dmtr(G) of trapezoid graphs” should be ”We have also established a new sharp relation between rads(G) and dmtr(G) of trapezoid graphs.”

We have modified the Subsection 1.3. We highlighted it by red colour.

(7.36) ”Based on our studied results, we design an optimal algorithms for finding central node(s), radius and diameter of trapezoid graph” should be ”Based on our studied results, we designed optimal algorithms for finding central node(s), radius, and diameter of trapezoid graphs.”

We have replaced ”Based on our studied results, we design an optimal algorithms for finding central node(s), radius and diameter of trapezoid graph” by ”Based on our studied results, we design an optimal algorithm for finding central node(s), radius and diameter of trapezoid graph”