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Vol. 9 Issue 8, August -2019 ISSN (O): 2249-3905, ISSN(P): 2349-6525 | Impact Factor: 7.196 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, Lamitan Extension Basilan 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, Lantawan and Akbar Municipalities. If public transportation is considered, the province needs at least 4 BSS or EA to be placed in Isabela City, Maluso 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 Zamboanga Peninsula (Region IX) and is bounded on the north by Basilan Strait, on the east by Moro Gulf, on the southeast by the Celebes Sea and on the west by the Sulu 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 Mindanao (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, Sumisip, 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.
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