International Journal of Research in Engineering and Applied Sciences(IJREAS) Vol. 9 Issue 8, August -2019 ISSN (O): 2249-3905, ISSN(P): 2349-6525 | Impact Factor: 7.196

Practical Application of Some Domination in Graphs

Emerson A. Perez1, Mathematics Department, Extension State College, Lamitan City, Basilan

Benjier H. Arriola2, Mathematics Department, College of Education Basilan State College, Isabela City, Basilan

Abstract

This paper shows a specific practical application of domination theories with a certain parameter that it is possible to construct a graph model to visualize or analyze with vertex as the location of places and edges as the connection between them. Variants of domination can be applied depending on the types or kinds of the graph being constructed. For the province of Basilan with consideration of placing basic social services (BSS) or economic activities (EA) in any of the municipalities or cities, there are concepts that can be directly used to analyze for a given parameter and these are the concepts of domination, independent domination, and secure domination. It was shown that with a given access road, the province needs at least five (5) basic social services or economic activities to be placed in Tipo-Tipo, Tabuanlasa, Hji. Muhtamad, and Akbar Municipalities. If public transportation is considered, the province needs at least 4 BSS or EA to be placed in Isabela City, Municipality, Tipo-Tipo Municipality, and Akbar Municipality. It is also possible to construct a graph model having access road as a connection between any two places with at least three strategic locations of BSS or EA which can be placed in Hji. Muhtamad, Tabuanlasa and Tipo-Tipo Municipalities. Considering the public transportation through either by sea or land, it is possible to construct a graph model with at least one strategic location of basic social services or economic activities which can be placed in Lamitan City.

Keywords: Domination, Independent Domination, Secure Domination, graph application

INTRODUCTION

Graph Theory began when Euler (1707-1781) solved a famous unsolved problem in 1736 called the Konigsberg Bridge Problem. There were two islands linked to each other and to the banks of the Pregel River by seven bridges. The problem was to begin at any of the four land areas, walk across each bridge once and return to the starting point. In proving that the problem is unsolvable, Euler replaced each land area by a point and each bridge by a line joining the corresponding points, thereby producing a "graph". Each point was labeled to correspond to the four land areas and proving that the problem is unsolvable by showing that the graph cannot be traversed in a certain way (Harrary, 1969).

Graph Theory became famous due to its application in computer science, engineering, physics and other fields of sciences. One of the notable concepts in this field is the theory of domination. The notion began as a partition problem, wherein the idea is to partition the set into two or more sets such that one set has a relationship to the other sets. Haynes, et. al. (1998) listed in their survey around 1,200 papers related to domination in a graph. The idea of domination may considerably be associated with roads as intended for the internal circulations.

Dharwadker & Pirzada (2007) studied graph theory to illicitly solve DNA sequencing using minimum vertex covers to removes the fewest possible Single Nucleotide Polymorphisms (SNPs) that will eliminate all conflicts. Filiol (2007), as cited by Dharwadker & Pirzada (2007), used the vertex cover algorithm to simulate the propagation of stealth worms on large computer networks and design optimal strategies for protecting the network against such virus attacks in real-time. Moreover, Dharwadker & Pirzada (2007) shows that it is

International Journal of Research in Engineering & Applied Sciences Email:- [email protected], http://www.euroasiapub.org 1 An open access scholarly, Online, print, peer-reviewed, interdisciplinary, monthly, and fully refereed journal. International Journal of Research in Engineering and Applied Sciences(IJREAS) Vol. 9 Issue 8, August - 2019 ISSN (O): 2249-3905, ISSN(P): 2349-6525 | Impact Factor: 7.196

possible to color the map of India using only four colors such that no two adjacent regions are assigned the same color.

The concept of graph theory can simply be applied to solve a certain situation that can be represented as a set on which relationship among its elements serves as their connection. Particularly, maps and network can be represented by the graph where the vertex represents a position and an edge represents a connection or relationship.

Let us consider a situation on which locations serve as vertices and roads serve as edges. Specifically, let us consider the Map of Basilan Province. This province is located across the southern tip of Peninsula (Region IX) and is bounded on the north by Basilan Strait, on the east by Moro Gulf, on the southeast by the and on the west by the Sea. Geographically, it lies between latitudes 6o16'48" and 6o45'56" north and between longitudes 121o26'00" and 122o24'38" east. The province of Basilan is one of the island provinces of the Autonomous Region in Muslim (ARMM) and is separated from the mainland of Mindanao by a strait of about 17 miles at its narrowest margin. The province is composed of two-component cities: Isabela and Lamitan, and eleven municipalities namely: Tuburan, Akbar, Moh. Ajul, Tipo-Tipo, Al-Barka, UnkayaPukan, , Tabuan-Lasa, Maluso, Lantawan and Hji. Mutamad. The Basilan Circumferential Road has a total length of 156.735 kilometers while roads under the jurisdiction of the Provincial Government have a total length of 236.30 kilometers.

Figure 1: Map of Basilan Province

Let us defined access road as the road connecting two locations or communities that can be possible to transport from one location to another via land vehicles such as truck, jeep, or tricycle. Public transportation is a possible distance between two locations that can be traversed by the usual utility vehicle or boat either through land or sea. This paper intended to identify specific variants of domination theory applicable to these situations or graph and tried to determine what type of graphs that can be developed for Basilan in order to be cost- effective and be practical in terms of basic social services (BSS) and economic activities (EA).

The study is imperative considering the key elements that affect the communication among the people in terms of economic activities and other social services of Basilan. In many cases, strategies involving road projects provide essential opportunities in allowing and catalyzing expansion and progress in the Province. This

International Journal of Research in Engineering & Applied Sciences Email:- [email protected], http://www.euroasiapub.org 2 An open access scholarly, Online, print, peer-reviewed, interdisciplinary, monthly, and fully refereed journal. International Journal of Research in Engineering and Applied Sciences(IJREAS) Vol. 9 Issue 8, August - 2019 ISSN (O): 2249-3905, ISSN(P): 2349-6525 | Impact Factor: 7.196

study provides relevant impact which would give an idea to be cost-effective in dealing with the resources of the government.

Moreover, the result of this paper may assist either the national or local government to strategize in establishing important facilities into its proper position for the province of Basilan. The graph model that can be developed in this study may also serve as a guide for some local government officials in planning provincial development Likewise, this study can also contribute in understanding, formalizing and applying mathematical models in some areas of research.

BASIC CONCEPTS AND PRELIMINARY NOTIONS

This section contains some of the basic concepts in Graph Theory as well as those concepts which are considered in this study.

Definition 1. Let 퐺 be a graph. A subset푆 is called dominating set of 퐺 if for every vertex 푢 ∈ 푉 ∖ 푆, there exist a vertex 푢 ∈ 푆 such that 푢 and 푣 are neighbors in 퐺. The domination number of 퐺, denoted by 훾(퐺),is the smallest cardinality of dominating set of 퐺 (Haynes et al, 1998).

Example 1. Let 푆 = {푢1, 푢2, 푢4}be the set of vertices of the graph shown in Figure 4. Then 푆 = {푢3, 푢5, 푢6}. Observed that 푢3 has a neighbor 푢4 ∈ 푆, 푢5 has a neighbor in 푢2 ∈ 푆and 푢6 has a neighbor 푢1 ∈ 푆, so that the set ∗ 푆 is a dominating set of 퐺. Similarly, the set 푆 = {푢2, 푢3, 푢5, 푢6}is also dominating set of 퐺. Since S has a minimum cardinality and there is no dominating set of cardinality less than 3, the domination number of G is 3. Thus, 훾(퐺) = 3.

푢1 푢2

푢5

푢6 푢3 푢4

Figure 2: Graph with Domination Number of 3

Definition 2. Let G be a graph. A dominating set S is called an independent dominating set of G if the set S is an independent set. The independent domination number of G, denoted by 훾푖(퐺), is the smallest cardinality of a connected dominating set of G (Haynes et al, 1998).

Definition 3. Let G be a graph. A dominating set S is called a secure dominating set of G if, for every vertex v in V \ G, there exists a vertex u in S such that (S \ {u} U {v}) is a dominating set of G. The secure domination number of G, denoted by훾푠(퐺), is the smallest cardinality of a secure dominating set of G (Cockayne et al, 2003).

Example 2. Let 푆 = {푢1, 푢2, 푢4} be the set of vertices of the graph shown in Figure 5. From the previous example, the set S is a dominating set. Now 푉 ∖ 푆 = {푢3, 푢5, 푢6, 푢7} and observed that 푢6 and 푢5 have the same neighbor in S which is 푢2. If we remove 푢2 from Sand interchange it with either 푢5or 푢7, the resulting set is not a dominating set of G and hence S is not a secure dominating set. If we add the vertex 푢5 to S, the resulting set becomes a secure dominating set and this set is the minimal secure dominating set of G. Thus, 훾s(퐺)=4. Note that the set S is not an ∗ independent set since 푢1 and 푢2 are a neighbor in S. Next, let us consider the set 푆 = {푢2, 푢3, 푢6}. Since none of the elements of S* are adjacent and the set S* is a dominating set, the set S* is an independent dominating set of G. Therefore, 훾i(퐺)=3.

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푢1 푢2

푢5

푢7

푢6 푢3 푢4

Figure 3: Graph with Vi(G) =3 and Vs(G)=4

MAIN RESULT

Basilan is divided into two (2) cities and eleven (11) municipalities (see Figure 1). In terms of the access road, that is, any vehicle is possible to pass through the road; the given map in Figure 1 can be converted into the graph as shown in Figure 4.

Isabela City

Hji. Muhtamad Akbar

Lantawan

Maluso Tuburan

Tipo-Tipo

Sumisip Al-Barka

Tabuanlasa

Figure 4: Graph of Basilan in terms of Access Roads

In Figure 4, the shaded vertices correspond to the location while the edges correspond to the existing road connecting two locations. Suppose a basic social service (BSS) will be constructed in some locations. Since the Tabuanlasa and Hji. Muhtamad are isolated locations, so they must have each a BSS. For the mainland, let us consider putting BSS to the location with highest neighbors, that is, putting it in Tipo-Tipo. Observe that there are five locations that are directly connected to Tipo-Tipo and these are Lamitan, Tuburan, Al-Barka, Ungkaya Pukan, and Sumisip. The next BSS will be placed in another location with the highest neighbor and this will be in Lamitan. The neighbors of Lamitan are Isabela, Akbar, and Tuburan. The only remaining places that have no direct access to the constructed BSS are Maluso and Lantawan. Since these two locations are neighbors, so we can choose either Maluso or Lantawan. Thus, the following locations will be the possible places where to put the BSS such as Tipo-Tipo, Lamitan, Lantawan, Hji. Muhtamad and Tabuanlasa. The concept being discussed is called domination in graph theory. Hence, in order to serve every municipalities or city, the range of number of BSS needed in Basilan is from five (5) to thirteen (13).

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Using the concept of domination, it is impossible to construct less than 5 BSS in Basilan with a given condition that any place will not pass through another place before accessing at least any BSS. To construct five (5) BSS in the province, there are only two combinations which are: Hji. Muhtamad, Tabuanlasa, Lantawan, Tipo- Tipo, or Hji. Muhtamad, Tabuanlasa, Lantawan, Tipo-Tipo, Akbar. Figure 5 shows the location of BSS in the shaded vertices.

Isabela City Isabela City

Hji. Muhtamad Hji. Muhtamad Akbar Akbar Lantawan Lantawan Ungkaya Pukan Ungkaya Pukan

Maluso Tuburan Maluso Tuburan

Tipo-Tipo Tipo-Tipo

Sumisip Sumisip Al-Barka Al-Barka

Tabuanlasa Tabuanlasa

Figure 5: Graph of Locations of BSS

Observe that, for the two combinations given, each location of BSS are not neighbor of any other municipalities/cities and mathematically called independent set. Hence, in order to have effective BSS with a minimal cost, we may construct exactly five (5) BSS in the mentioned locations.

Theorem 1. Let 퐺 be a graph. If 퐶1, 퐶2, 퐶3, … , 퐶푘 are components of 퐺 such that 〈푉(퐺)〉 ≅ 〈퐶1 ∪ 퐶2 ∪ 퐶3 ∪ … ∪ 퐶푘〉, then the following are true: (i.) 훾(퐺) = 훾(퐶1) + 훾(퐶2) + 훾(퐶3) + ⋯ + 훾(퐶푘); (ii.) 훾푠(퐺) = 훾푠(퐶1) + 훾푠(퐶2) + 훾푠(퐶3) + ⋯ + 훾푠(퐶푘); and (iii.) 훾푖(퐺) = 훾푖(퐶1) + 훾푖(퐶2) + 훾푖 (퐶3) + ⋯ + 훾푖(퐶푘)

Proof: If 퐺 is connected, then we are done. Suppose 퐺 is not connected and let 퐶1, 퐶2, 퐶3, … , 퐶푘 be components of 퐺. If 푆1, 푆2, 푆3, …, 푆푘 are dominating sets of 퐶1, 퐶2, 퐶3, … , 퐶푘, respectively, then 훾(퐺)=훾(〈퐶1 ∪ 퐶2 ∪ 퐶3 ∪ … ∪ 퐶푘〉) ≤ 훾(퐶1) + 훾(퐶2) + 훾(퐶3) + ⋯ + 훾(퐶푘). Let 푆 be the dominating set of 퐺. Since 푆1 ∩ 푆2 ∩ 푆3 ∩ … ∩ 푆푘 = ∅ from the fact that 퐶1, 퐶2, 퐶3, … , 퐶푘 are components of 퐺, and so 푆1 ∩ 푆2 ∩ 푆3 ∩ … ∩ 푆푘 ⊆ 푆. Thus, |푆1| + |푆2| + |푆3| + ⋯ + |푆푘| ⊆ |푆|. This implies that 훾(퐺) ≥ 훾(퐶1) + 훾(퐶2) + 훾(퐶3) + ⋯ + 훾(퐶푘). The proof for the secure and independent dominating sets are similar and straightforward. ∎

Example 3. Let G be the graph as shown in Figure 4. Let 퐶1= {Isabela, Lamitan, Akbar, Hji. Mohammad Ajul, Tuburan, Tipo-Tipo, Al-Barka, Ungkaya Pukan, Sumisip, Maluso, Lantawan}, 퐶2={Tabuanlasa} and 퐶3={Hji Muhtamad}. Clearly, 훾(퐶2) = 훾(퐶3) = 1 being both an isolated vertex. Note that the set {Tipo-Tipo, Akbar, Lantawan} is a dominating set of the subgraph 〈퐶1〉. Since there is no dominating set for 〈퐶1〉 with cardinality 2, and so 훾(〈퐶1〉) =3. Thus, 훾(퐺) = 훾(퐶1) + 훾(퐶2) + 훾(퐶3) = 5. Also, since 퐶1, 퐶2, and 퐶2 are all independent sets, so 훾푖(퐺) = 훾푖(퐶1) + 훾푖(퐶2) + 훾푖 (퐶3).

Next, let us consider the availability of the public transportation. That is, a public transportation is available from one location to another. The graph with available transportation is shown in Figure 6.

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Isabela City

Hji. Muhtamad Akbar

Lantawan Ungkaya Pukan Maluso Tuburan

Tipo-Tipo

Sumisip Al-Barka

Tabuanlasa

Figure 6: Graph with Public Transportation

Given the previous conditions, we want to construct BSS in some of the given locations such that the number of BSS will be the minimum. If we place BSS in Isabela, then Hji. Muhtamad, Lantawan, and Lamitan will be served. Since Tabuanlasa is a pendant vertex, another BSS must be constructed in Maluso. Sumisip now being served from BSS in Maluso and since there are two pendant vertices connected to Tipo-Tipo, BSS must also be constructed in Tipo-Tipo. Thus, every municipalities/city are being served except Akbar and Hadji Mohammad Ajul. Hence, we need to construct another BSS in either Akbar or Hadji Mohammad Ajul. Observe that if we want to serve all locations, then at least (4) four BSS must be constructed in different locations as mentioned. Hence, the following are the combinations of municipalities/cities that BSS must be constructed to minimize the number of locations without sacrificing the services needed. These are 1. Isabela, Maluso, Tipo-Tipo, Hadji Mohammad Ajul; or 2. Isabela, Maluso, Tipo-Tipo, and akbar. Similarly, the concept being discussed is called a dominating set of a graph.

Remark 1. Let G be the graph as shown in Figure 4. Then 훾(퐺) = 4.

Proof. Let A={Isabela, Maluso, Tipo-Tipo, Akbar} be the set of some municipalities/cities in Basilan. Note that N(A) ={Hji. Muhtamad, Lantawan, Lamitan, Hadji Mohammad Ajul, Tuburan, Ungkaya Pukan, Al-Barka, Sumisip, Tabuanlasa} and 퐴 ∪ 푁(퐴) = 푉(퐺), it follows that A is a dominating set of G. Let C be a dominating set of G such that |C|<|A|. Since Hji. Muhtamad, Tabuanlasa, Al-Barka, and UngkayaPukan, either these municipalities are in C or Isabela, Maluso and Tipo-Tipo are in C. Observed that the number of pendant vertices is 4 and thus cannot be equal to C. Hence, C={Isabela, Maluso, Tipo-Tipo}. All municipalities/cities are adjacent to any municipality in C except Akbar and Hadji Mohammad Ajul. Hence, there is no dominating set of order 3 which is a contradiction. Therefore, 훾(퐺) = 4. ■

If we consider the possibility of constructing an access road from one location to another location, then the minimum number of location for a BSS to be constructed so that everyone has access to these services either through walking or vehicle is three (3). These three locations are in Hji. Muhtamad, Tabuanlasa and any from the other municipalities or cities. Since Hji. Muhtamad and Tabuanlasa are fixed location for BSS, so let us choose a particular location from the remaining locations. Supposed, Isabela will be chosen, then new roads must be constructed from this location directly to nine (9) other locations such as Akbar, Hadji Mohammad Ajul, Tuburan, Tipo-Tipo, UngkayaPukan, Al-Barka, Maluso, and Sumisip (see Figure 7).

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Isabela City

Hji. Muhtamad Akbar

Lantawan Ungkaya Pukan

Maluso Tuburan

Tipo-Tipo

Sumisip Al-Barka

Tabuanlasa Figure 7: Location of BSS-Isabela

If Lamitan will be chosen, then new road must be constructed from this location directly to other six (6) locations such as Hadji Mohammad Ajul, Al-Barka, Ungkaya Pukan, Lantawan, Maluso, and Sumisip (see Figure 8).

Isabela City

Hji. Muhtamad Akbar

Lantawan

Ungkaya Pukan

Maluso Tuburan

Tipo-Tipo

Sumisip Al-Barka

Tabuanlasa

Figure 8: Location of BSS-Lamitan

If Lantawan will be chosen, then new road must be constructed from this location directly to eight (8) other locations such as Lamitan, Akbar, Hadji Mohammad Ajul, UngkayaPukan, Tipo-Tipo, Al-Barka, Tuburan, and Sumisip (see Figure 9).

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Isabela City

Hji. Muhtamad Akbar Lantawan

Ungkaya Pukan

Maluso Tuburan

Tipo-Tipo

Sumisip Al-Barka

Tabuanlasa

Figure 9: Location of BSS-Lantawan

Choosing exactly one location for BSS and the number of new roads constructed connecting the other locations is shown in Table 1.

Table 1-Possible Number of BSS can be constructed BSS Constructed Number of New Road to be Constructed Isabela 8 Lamitan 6 Akbar 7 Hadji Mohammad Ajul 8 Tuburan 7 UngkayaPukan 9 Tipo-Tipo 5 Al-Barka 9 Sumisip 8 Maluso 8 Lantawan 8

Without considering any distances, the Tipo-Tipo municipality is the strategic location for the BSS in order to be cost-effective and practical since it has lesser number of new roads to be constructed. See Figure 10 for the graph after new roads will be included.

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Figure 10: Location of BSS- Tipo-Tipo

Next, let us consider EA. Supposed we want to identify the location wherein the product and distribution of goods or services will be available in that location. If the only location is needed, then it is similar to the previous discussions. Supposed in addition, change of EA location will be allowed. That is, in case of changes in EA location such that after changes every location is still being served by the new set of EA. This concept is called secure domination. Let us consider the set {Tipo-Tipo, Tabuanlasa, Hji. Muhtamad, Lantawan, Akbar} and observed that if EA in Tipo-Tipo will be moved to UngkayaPukan, there is a municipality that cannot be served. Thus, we need to include UngkayaPukan in the set. Following the concept of secure domination of the set {Hji.Muhtamad, Tabuanlasa, UngkayaPukan, Tipo-Tipo, Lamitan, Akbar, Lantawan, Maluso}. This set is the minimum number of combinations that satisfies the above condition. The specific concept applicable to this situation is called secure domination.

Remark 2. Let G be the graph shown in Figure 9. Then 훾푠(퐺) = 8.

Proof. Let A={Hji.Muhtamad, Tabuanlasa, UngkayaPukan, Tipo-Tipo, Lamitan, Akbar, Lantawan, Maluso} be the secure dominating set of G. Now V (G) \ A={Isabela, Hadji Mohammad Ajul, Tuburan, Al-Barka, Sumisip}. Note that Isabela is adjacent to Lamitan and Lantawan, and hence any EA from these two locations can be moved to Isabela. If EA from Lamitan will be moved to Isabela, Lamitan is still being served by EA in Akbar or Tipo-Tipo. Similarly, interchanging the EA location between Hadji Mohammad Ajul and Akbar, the new set is still a dominating set. Interchanging Tuburan with Lamitan, Al-Barka with Tipo-Tipo, and Sumisip with Maluso, the new set is always a dominating set. Hence, the set A is a secure dominating set of G. Subtracting any one municipality or city in the set A makes it a non-secure dominating set. Therefore, 훾푠(퐺) = 8. ■

Supposed another EA will be considered by putting it in a place like seaport in that particular location. Let us consider the present situation in Basilan wherein seaports (big sea vessel can dock) are available in Isabela, Lamitan, and Maluso. Observed that almost all places are being served by any of these ports, except Hadji Mohammad Ajul, Unkaya Pukan, and Al-Barka. Since Ungkaya Pukan is adjacent to Tipo-Tipo only and this place has no seashore were to construct the public seaport, then the combination is not possible with a condition that every municipality/city has access to the latter directly without passing through the other municipality/city.

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Now, let us consider public transportation (either by land or by sea) from any place to another. Suppose we want to place exactly one BSS/EA in any location so that every municipalities/city have access to the said BSS/EA. Let us start constructing BSS/EA in Isabela such that all other locations could have access to the BSS/EA directly without passing any other locations. Then new roads must be constructed from Isabela to Maluso, Ungkaya Pukan, Sumisip, Akbar, Hji. Mohammad Ajul, Tuburan, Tipo-tipo, and Al-Barka. Public transportation must also be initiated from Isabela to Tabuanlasa. Similarly, if we put in place the BSS/EA in other locations, we need to construct new roads or initiate public transportations from connecting the other locations directly from the identified locations. The number of either new road to be constructed or public transportation to be initiated is summarized in Table I. Observed that for some locations, it is not possible to have public transportation directly without passing from another municipalities/cities. If we consider the minimum number of public transportation, Lamitan has eight (8) public transportations and hence will be the strategic location for BSS or EA. The graph of this situation showing the direction of public transportation is given in Figure 11.

Figure 11: Location for BSS or EA in Lamitan

CONCLUSION

There are two graphs that can be constructed using the map of Basilan. In terms of access road, we have shown that there are two isolated locations and there are thirteen (13) edges (access road) from different municipalities/cities as shown in Figure 4. In terms of distance with passable transportation there are fifteen (15) edges connecting some of the municipalities/cities as shown in Figure 6. If BSS will be considered, there are two specific variants of domination were used and these are the dominating set and independent dominating set. For EA, there were two variants of domination were used and these are dominating set and secure dominating set.

Considering the concept of dominating set, in order to be cost-effective and practical in terms of number of BSS to be constructed, Figure 4 shows the graph with locations of Basic Social Services. Example 3 states that to minimize the number of BSS to be constructed, the province needs at least five (5) locations so that every constituent has access to these BSS by walking or vehicle. For the availability of public transportation, Remark 1 states that there is only four (4) location needed to minimize the number of locations. In terms of access roads, if we need to construct one location in the mainland so that every other municipalities/city have access to it, Tipo-Tipo is the strategic location as shown in Figure 10. In terms of public transportation, if we only need one strategic location such that it is accessible to all municipalities/cities either by sea or land, then Lamitan is the strategic location as shown in Figure 11.

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