Ratio Mathematica Volume 44,2022 Product Signed Domination in Graphs T. M. Velammal * A. Nagarajan † K. Palani ‡ Abstract Let be a simple graph. The closed neighborhood of , denoted by , is the set . A function is a product signed dominating function, if for every vertex where . The weight of , denoted by , is the sum of the function values of all the vertices in . . The product signed domination number of is the minimum positive weight of a product signed dominating function. In this paper, we establish bounds on the product signed domination number and estimate product signed domination number for some standard graphs. Keywords: graphs, product signed dominating function, product signed domination number. 2010 AMS subject classification: 05C69 § . * Research Scholar (Reg. No. 21212232092010), PG & Research Department of Mathematics, V.O. Chidambaram College, Thoothukudi-628008, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.avk.0912@gmail.com † Associate Professor (Retd.), V.O. Chidambaram College, Thoothukudi-628008, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.nagarajan.voc@gmail.com ‡ Associate Professor, A.P.C. Mahalaxmi College For Women, Thoothukudi-628002, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India. palani@apcmcollege.ac.in § Received on June 8th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.923. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement 340 mailto:avk.0912@gmail.com mailto:nagarajan.voc@gmail.com mailto:palani@apcmcollege.ac.in T. M. Velammal, A. Nagarajan, and K. Palani 1. Introduction The fundamental thought of graphs was first presented in eighteenth era by Swiss Mathematician Leonhard Euler. It has numerous applications in Natural Sciences, Technology, Information System Research and so on. The quickest developing region in theory of graph is domination. Ore introduced the terms “Dominating Set” and “Domination Number”. Dunbar et al. introduced signed domination number [1],[2],[4],[5]. Hosseini gave a lower and upper bound for the signed domination number of any graph [3]. In this paper, we introduce the concept of product signed domination number and find bounds on product signed domination number. 2. Preliminaries Definition 2.1: A comb graph is a graph obtained by joining a pendant edge to each vertex of a path. Definition 2.2: A star graph is a tree on vertices with one vertex having degree and the other vertices having degree . Definition 2.3: A tree containing exactly two non-pendant vertices is called a double star. It is denoted by 3. Main Results Definition 3.1: Let be a simple graph. The closed neighborhood of , denoted by , is the set . A function is a product signed dominating function, if for every vertex where . The weight of , denoted by , is the sum of the function values of all the vertices in . The product signed domination number of is the minimum positive weight of a product signed dominating function. Observation 3.2. (i) In a graph , a pendant vertex and its corresponding support vertex get the same functional values (i.e.) either or since otherwise . (ii) In a product signed dominating function, all the vertices of a graph should not be assigned since product signed domination number is positive. 341 Product Signed Domination in Graphs (iii) In a product signed dominating function, for every vertex contains either zero or even number of vertices with functional value , since otherwise . (iv) If and denote the number of vertices with functional values and respectively, then . Theorem 3.3: the total number of vertices. Proof: Let be a complete graph on vertices. Let and . Since each pair of vertices is connected by an edge, in a product signed dominating function the number of vertices with functional value must be even. Case 1: Define a function as follows. When , every vertex should be assigned under , since otherwise would not be a product signed dominating function with a positive weight. the total number of vertices. Case 2: and is odd Subcase 2.1: is even Partition the vertex set into two sets and such that , and . Define as Obviously, for every , and hence is a product signed dominating function. 342 T. M. Velammal, A. Nagarajan, and K. Palani Subcase 2.2. is odd Partition the vertex set into two sets and such that , and . Here is odd and is even. Define as Clearly, for every , .Hence is a product signed dominating function. Also, . Since is odd. This function gives the minimum value for product signed domination number. Case 3: and is even Subcase 3.1: is even If we partition the vertex set into two sets and such that , and and assign to all the vertices in and to all the vertices in , then the function would be a product signed dominating function but the weight would be zero. Since is even, is odd. Partition the vertex set into two sets and such that , and . Define as Therefore is a product signed dominating function. Also . Since is even, this function gives the minimum value for product signed domination number as before. Subcase 3.2: is odd We have is even. Partition the vertex set into two sets and such that , and . Define as Correspondingly, for every , . Hence is a product signed dominating function. Also . Proceeding as above, this function gives the minimum value for product signed domination number. 343 Product Signed Domination in Graphs the total number of vertices. Theorem 3.4: For the comb graph, , the product signed domination number , the total number of vertices. Proof: Let be a comb graph . Let be the vertex set with ’s representing the pendant vertices and be the edge set. Since is the pendant vertex to , both and must be either or for (by 3.2(i)). Case 1: is odd Define as follows. If and are both assigned , then and should be assigned since otherwise would be . Further if , again . Hence , , , and so on. Then and is a product signed dominating function. Correspondingly, the weight of the graph is a negative integer which is a contradiction to the weight is positive. Hence, let us start with . Then , since otherwise is not a product signed dominating function. Hence is the only product signed dominating function having a positive weight. Hence it is the unique product signed dominating function. the total number of vertices of Case 2: is even Define as follows. If and are both assigned , then and should be assigned since otherwise would be . Further if , again . Hence 344 T. M. Velammal, A. Nagarajan, and K. Palani , , , and so on. Then and hence . Therefore, this is not a product signed dominating function. Hence, let us start with . Then , since otherwise is not a product signed dominating function. Hence is the unique product signed dominating function. the total number of vertices of By cases 1 and 2, . Observation 3.5: For any graph , total number of vertices of . Here the bounds are sharp since and total number of vertices. Theorem 3.6: The product signed domination number of a path on vertices is equal to . Proof: Let be a path on vertices. If Then (or) If Then (or) By the above observation, if is assigned , then must be assigned so that . Then must be assigned so that . So must be assigned so that . Proceeding like this, we define a function as follows. For So when is not a product signed dominating function since When is not a product signed dominating function since When is a product signed dominating function having a negative weight. So let us try with assigned to . If is assigned , must be assigned so that . Again must be assigned so that . Again must be assigned so that and so on. Therefore, . And is a minimum positive weight product signed dominating function. 345 Product Signed Domination in Graphs The weight of this function , the total number of vertices. Therefore, , the total number of vertices. Theorem 3.7: The product signed domination number of a cycle on vertices is equal to . Proof: Let be a path on vertices. If Then (or) If Then (or) By the above observation, if is assigned , then must be assigned so that . Then must be assigned so that . So must be assigned so that . Proceeding like this, we define a function as follows. For So, when is a product signed dominating function having negative weight. When is not a product signed dominating function since . When is not a product signed dominating function since . So let us try with assigned to . If is assigned , must be assigned so that . Again must be assigned so that . Again must be assigned so that and so on. Therefore, . And is a minimum positive weight product signed dominating function. The weight of this function , the total number of vertices. Therefore, , the total number of vertices. Theorem 3.8: The product signed domination number of a star graph on vertices is equal to . Proof: Let be a star graph on vertices. Let and . By 3.2(i), and should be assigned same functional value. 346 T. M. Velammal, A. Nagarajan, and K. Palani If , then the weight of is negative. Therefore must be equal to and hence define as . And obviously is the minimum positive weight product signed dominating function. Therefore, , the total number of vertices. Theorem 3.9: The product signed domination number of a double star graph is equal to . Proof: Let be a double star graph on vertices. Let and . Case 1: Number of pendant vertices to atleast one of is odd. Without loss of generality, assume that number of pendant vertices to is odd. If we assign to , then all the pendant vertices to must be assigned (by 3.2(i)). Since number of pendant vertices to is odd, must be assigned . Hence again by 3.2(i), all the pendant vertices to get . But here . So this is not a product signed dominating function. Hence define as Clearly, is the minimum positive weight product signed dominating function. Therefore, , the total number of vertices. Case 2: Number of pendant vertices to both and is even. If we assign to , then all the pendant vertices to must be assigned (by 3.2(i)). Since number of pendant vertices to is even, must be assigned . Hence again by 3.2(i), the pendant vertices to get . Here this is a product signed dominating function having a negative weight. So, the only possible positive weight product signed dominating function is Therefore, , the total number of vertices. 347 Product Signed Domination in Graphs References [1] J. Dunbar, S.T. Hedetniemi. Henning, and P.J. 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