Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 371 https://internationalpubls.com Some Bench Mark Results on Total Domination Subdivision Stable Graph A. Jeeva1*, M. Yamuna2, A. Kuppan3, V. Sivan4, P. Selvaraju5, K. Kalpana6 1Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and Technology, Chennai, Tamil Nadu, India. drjeevaa@veltech.edu.in 2Department of Mathematics, School of Advanced Sciences, Vellore Institute of Science and Technology, Tamil Nadu, India. myamuna@vit.ac.in 3Department of Mathematics, Saveetha Engineering College, Chennai, Tamilnadu, India. kuppanmyname@gmail.com 4Department of Mathematics, Saveetha Engineering College, Chennai, Tamilnadu, India. shivan.ve@gmail.com 5Department of computer science and Engineering, Saveetha School Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, Tamilnadu, India pselvar@yahoo.com 6Department of Chemistry, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, Tamilnadu, India kalpanakumar19@gmail.com *1Corresponding author Email id: drjeevaa@veltech.edu.in Article History: Received: 20-04-2024 Revised: 10-06-2024 Accepted: 23-06-2024 Abstract: For a graph G, the total dominating set defined as a set of vertices in S such that all the vertices in V(G) has at least one neighbor in S, the least cardinality is noted as t(G). The total domination number of each and every graph while subdividing any edge xy of G is equal to the total domination number of G, which results in the total domination subdivision stable graph abbreviated as TDSS and the symbolic expression is Gtsd(xy). The research paper, we introduce TDSS and proposed conditions under which a graph is TDSS and not TDSS. Keywords: Total domination, Total domination subdivision, Total domination subdivision stable (TDSS). 1. Introduction All graphs considered here simple, connected and undirected graph with V and E which follows vertex set and edge set. For basic terminology and notations for graphs and domination parameters which is not defined here refer [1] and [2] respectively. The boundary of D defined [2] as B (D) = N ( D ) – D. Let x  G, the vertex x is called good [3] such that if all possible t – sets contained the vertex x otherwise it is called bad vertex. If a vertex x is needed only to dominate itself in the minimum dominating set then x is called selfish. Any vertex we call t-dominated in V – D at least t vertices needed to dominate that vertex. If, after removing a vertex x from G, we attain the graph G – Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 372 https://internationalpubls.com x, which is less than the total domination number of G, that is, t(G – x)  t(G), then that vertex x is considered down. The total domination subdivision number was first investigated in [4]. The total domination subdivision number increased by subdividing least number of edges of G which is discussed in [5]. It has been studied by several authors in [6,7,8,9 & 10]. The authors [11] introduced the graph named as domination subdivision stable graph is the domination number of each and every graph while subdividing any edge of G is equal to the domination number of G, which is abbreviated as DSS. Let e = xy be an edge with end points {x, y} of G. While subdividing it, we access a new one say z and having new edges say {x, z} and {z, y} of the resulting graph, it is expressed as G sd xy. Based on this concept, we extend this to total domination and introduce a new graph named as total domination subdivision stable graph. 2. Main Results The focus of this section, we defined Total domination subdivision stable graph and discussed basic properties for obtaining TDSS from a graph G. Definition 2.1 For a given graph G, the total domination number of all graphs derived by subdividing any edge xy of G is same for the total domination number of G which named as total domination subdivision stable. Using this graph operation (subdivision) of any edge xy, we obtain a new vertex z, the expression of this denoted Gtsd xy = z. Fig. 1 Graph G and Gtsd 14 In Fig 1,  t(G) = t(G tsd 14) = 4. According to the Fig 1, the total domination number is the same for every pair. Therefore t ( G tsd xy ) =  t ( G ). Theorem 2.2 For every graph G, t (G tsd xy) ≥ t (G)  xy  E (G). Proof Let us assume the graph to be G and t – set of G to be D. Consider G tsd xy where e = xy  E (G) and assume D1 to be a t – set for G tsd xy. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 373 https://internationalpubls.com If possible let | D1 | < | D |. Case 1 z  D1 In this case the possible conditions are either, 1. x, y  D1 or 2. x or y  D1. Let x, y  D1 ie., x, y, z  D1 , then we obtain D2 where D2 = D1 – {z} is a t – set for G so | D2 | < | D |, we get a contradiction. If either x or y  D1 say x  D1, then we obtain D3 where D3 = D1 – {z}  {y} as a t – set for G such that| D3| < | D |, and we get a contradiction. Case 2 z  D1 In this case the possible conditions are either, 1. x, y  D1 or 2. x or y  D1. In both cases, we get a contradiction for a t – set D1. Since for G, D1 itself is a t – set we get | D1 | < | D |. Thus t (Gtsd xy) ≥ t (G)  e = xy in E (G).  Theorem 2.3 For a given graph G such that t (G) = 2t (G), then G is TDSS graph Proof Assume that a total 2 – dominated graph is G and the t – set for G is D. Let e = xy  E (G). Case 1 x, y in D Let Gtsd xy = z. Then we get a t – set for Gtsd xy is D – {x}  {z} ie., t (Gtsd xy) = t (G). Case 2 x not in D, y in D Claim If y is a 2 – dominated vertex such that y is adjacent to x, z where x, z  t (G), then we get t (Gtsd xy) = t (G) and also t (Gtsd zy) = t (G). Proof Let G be any graph and the 2- dominated vertex is x. Let Gtsd xy = s. In Gtsd xy, vertex x dominates s and z dominates y. Hence t (Gtsd xy) = t (G). Similarly t (Gtsd zy) = t (G). By the above claim, t (Gtsd xy) = t (G). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 374 https://internationalpubls.com Case 3 x in D, y not in D Similarly if x in D and y not in D, we get t (Gtsd xy) = t (G). This is true  e = xy  E (G). Therefore, G is TDSS.  Converse need not be true. Example Fig. 2 TDSS graph In Fig.2, G is TDSS but x is not a 2 – dominated vertex. Corollary If G is a graph with x, y  V (G),  a t – set D where x is adjacent to y, x  D and y is 2 – dominated then G is TDSS. Proof Let x, y  V (G) and t – set for G be D such that x  D and y is 2 – dominated. Let Gtsd xy = z. By Theorem 2.3, we have D is a t – set for Gtsd xy. It is true for remaining x, y  V (G). Therefore, G is TDSS.  Example Fig. 3 Graph G In Fig. 3, the vertices x and y are such that x  t (G) and y is 2 – dominated for the t – set of G. Therefore G is TDSS. Remark If every vertex is 2 – dominated in V – D, then it follows the above corollary. Theorem 2.4 For every vertex is 2 – dominated in V – D  t – set of G then, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 375 https://internationalpubls.com (1) G does not have any pendant vertex. (2) If for x, z  t (G), x is a down vertex and x adjacent to z. Then  one s which is not adjacent to x but s is adjacent to z. Proof Let G be a given graph in which all vertices in V – D is 2 – dominated for all the t – set of G. (1) Assume that y is a pendant vertex of G. Since all the vertices in V – D is 2 – dominated, this implies that every pendant vertex must be included in every t – set of G. Consider x be a support vertex of y where x is neighbor of y. Therefore,  atleast one z such that z ≠ y, z  D, z  N (x). Then, we have a t – set D1 = D – {y}  {z} such that y is single dominated, we get a contradiction. Hence, D does not have any pendant vertex, ie.,  x  V (G), N (x) ≥ 2. (2) For each and every x  t (G), Pn [x, D] = . Let us consider x, z such that x is adjacent to z that belongs to t (G). Consider the graph G – x, we have the t – set for G – x is t (G) – {z} – {x}  {s} where s  N (z). Therefore, we get t (G – x) = t (G) – 1. Thus x is a down vertex if x  t (G).  Remark 1. By theorem 2.3 and 2.4 we see that if G is a TDSS graph such that every t – set of G is 2 – dominating, then every t – set of G includes all pendant and support vertices. 2. If x is a t – dominated vertex say y adjacent to y1, y2, …, yt where y1, y2, …, yt  t (G),t (Gtsd xy1) = t (Gtsd xy2) = . . . . = t (Gtsd xyk) = t (G) ie., a t – dominated graph is TDSS. Theorem 2.5 If either x or y is selfish and t (G) = 2 such that x, y  t (G) then G is TDSS. Proof Let x, y  t (G), the vertex y is selfish. Let Gtsd xs = t for some s  V (G). Then we have a t – set for Gtsd xs as t (G) – {y}  {t}.Therefore, G is TDSS.  Converse need not be true. Example Fig. 4 TDSS Graph with neither x nor y is selfish Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 376 https://internationalpubls.com In Fig.4, neither x nor y is selfish but G is a TDSS graph. Theorem 2.6 Let G be a graph with x  V (G). Let t – set for G is D such that x, y  D, x adjacent to y where y is selfish, B ({x, y})  D = . If this is possible for each and every vertices of G, then G is a TDSS. Proof Let G be a graph with x V (G) and t – set for G is D. By given condition  one y such that x, y  D, where y is selfish, x adjacent to y and B ({x, y})  D =  . Let Gtsd xs = z where s  N (x). Then we have t – set for Gtsd xs is D1 = D – {y}  {z}. That is true for each and every x  V (G) of G. Therefore G is TDSS.  Corollary For a graph G such that  x, y  V (G), there exists a t – set D such that B ({x, y})  D =  and either x or y is selfish. Then, G is TDSS. Proof Let G be a graph with x, y  V (G) such that x adjacent to y. Given that  one t – set D where x, y  D, such that B ({x, y})  D =  and either x or y is selfish. Let us assume that x is selfish and Gtsd xy = z. Then we have t – set for Gtsd xy is D – {x}  {z}. By theorem 2.5, this is true for all x, y  V (G). Therefore G is TDSS.  Converse need not be true. Example Fig. 5 TDSS graph with B ({x, y})  D   In Fig.5, we have B ({x, y})  D  . Then G is TDSS with x, y  D, x adjacent to y such that y is selfish. Theorem 2.7 If G is TDSS, then every pendant vertex of G is included in some t – set. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 377 https://internationalpubls.com Proof Let G be TDSS with x, y  V (G), here the pendant vertex is y and support vertex is x. Let Gtsd xy = z. In Gtsd xy, either y  t (Gtsd xy) or y  t (Gtsd xy). If y  t (Gtsd xy), then z  t (Gtsd xy), x   t (Gtsd xy). Here, the t – set for G containing y is t (Gtsd xy) – {z}  {x}. If y   t (Gtsd xy), then x, z  t (Gtsd xy). Here also the t – set for G containing y is t (Gtsd xy) – {z}  {y}. In all cases, there is a t – set containing y.  Corollary If G has at least one pendant vertex that is TDSS. Then G has at least one selfish vertex. 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