Ratio Mathematica Volume 44, 2022 Outer independent square free detour number of a graph K. Christy Rani * G. Priscilla Pacifica † Abstract For a connected graph , a set of vertices is called an outer independent square free detour set if is a square free detour set of such that either or is an independent set. The minimum cardinality of an outer independent square free detour set of is called an outer independent square free detour number of and is denoted by We determine the outer independent square free detour number of some graphs. We characterize the graph which realizes the result that for any pair of integers and with there exists a connected graph of order with square free detour number and outer independent square free detour number Keywords: square free detour set; outer independent square free detour set; outer independent square free detour number. 2010 subject classification: 05C12, 05C38 ‡ * Research Scholar, Reg. No.: 20122212092002, PG and Research Department of Mathematics, St. Mary’s College (Autonomous), Thoothukudi-628 001, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627 012, India; christy.agnes@gmail.com. † Assistant Professor, Department of Mathematics, St. Mary’s College (Autonomous), Thoothukudi- 628001, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, India; priscillamelwyn@gmail.com. ‡ Received on June 14th, 2022. Accepted on Sep 1st, 2022. Published on Nov30th, 2022. doi: 10.23755/rm.v44i0.919. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 309 K. Christy Rani and G. Priscilla Pacifica 1. Introduction In this article, a graph is considered to be a finite, undirected and connected graph of order with neither loops nor multiple edges. Let be the longest path in and a path of is called detour. The parameters on detour concept were developed by Chartrand [2]. The detour concept was extended to triangle free detour concept by S. Athisayanathan et al. [1, 7]. The detour concept was applied in domination by number of authors. The detour domination number was studied and extended to outer independent detour domination by number of authors in [5, 6]. For any two vertices in a connected graph , the path P is called triangle free path if no three vertices of P induce a triangle. The triangle free detour distance ( ) is the length of a longest triangle free path in A path of length ( ) is called a triangle free detour. A set of is called a triangle free detour set of if every vertex of lies on a triangle free detour joining a pair of vertices of . The triangle free detour number of is the minimum order of its triangle free detour sets. This triangle free detour number was studied by S. Athisayanathan and S. Sethu Ramalingam in [8]. This concept was extended to square free detour number by K. Christy Rani and G. Priscilla Pacifica [4]. A square free detour number of denoted by is defined as the minimum order of square free detour set consisting of every pair of vertices of all the square free detours in which every vertex of lies on. In this article, we introduce the outer independent square free detour number denoted by The outer independent square free detour number of some standard graphs and cycle related graphs are determined. For the basic terminologies we refer to Chartrand [2]. 2. Preliminaries The following theorems are used in the sequel. Theorem 2.1 [3] For any connected graph Theorem 2.2 [3] Every end-vertex of a non-trivial connected graph belongs to every detour set of . Theorem 2.3 [3] If T is a tree with end-vertices, then Theorem 2.4 [4] If is the cycle ), then 310 Outer independent square free detour number of a graph 3. Outer independent square free detour number of a graph Definition 3.1 Let be a simple connected graph of order A set of vertices is called an outer independent square free detour set in if is a square free detour set such that either or is independent. The minimum cardinality of an outer independent square free detour number of is called outer connected square free detour number of and is denoted by Example 3.2 For the graph shown in Figure 1, the set is a minimum outer independent square free detour set and is a minimum square free detour set for and so and Here we find that Moreover, the sets and are also the minimum outer independent square free detour sets of Hence there can be more than one minimum outer independent square free detour set for a graph Figure 1: Theorem 3.3 For any connected graph every end-vertex of belongs to every outer independent detour set of . Proof. Since every outer independent square free detour set is also a detour set of , the proof follows from Theorem 2.2. Theorem 3.4 For any connected graph Proof. The result follows from Theorems 2.1 and 3.3. Remark 3.5 The bounds in Theorem 3.4 are sharp. The set of two end-vertices of a path is its minimum outer independent square free detour set so that The bounds in Theorem 3.4 are also strict. For the graph of order 11 given in Figure 1, 311 K. Christy Rani and G. Priscilla Pacifica Theorem 3.6 If is a Path , then Proof. Let be a path of order and be a set of vertices. Then we consider two cases. Case 1. Let be odd. Let be the square free detour set such that is independent. Hence is an outer independent square free detour set. Thus Case 2. Let be even. Let be a square free detour set such that is independent. Thus The following corollary is immediate. Corollary 3.7 For any connected graph if and only if Theorem 3.8 If is a star , then Proof. Let be a star with end-vertices. Then by Theorem 3.3, where are the end-vertices of such that is independent. Hence By Theorem 2.3, is also a minimum detour set and so Hence Theorem 3.9 If is a complete bipartite graph , then Proof. Let be a complete bipartite graph of order with two partitions and where and Let be a set of vertices of Now, it is easy to verify that is independent. Hence . Remark 3.10 Due to the connectivity of the complete graph it is not possible to find the outer independent square free detour number for . Theorem 3.11 If is a cycle , then . Proof. Let be a cycle of order . Let be any set of vertices of . we consider two cases. Case 1. Let be odd. Let be a square free detour set such that is independent. Hence . Case 2. Let be even. Let be a square free detour set such that is independent. Hence . 312 Outer independent square free detour number of a graph From the above cases, we observe that . Then by Theorem 2.4, it follows that . Theorem 3.12 If is a Wheel, then . Proof. Let be a Wheel of order . Then is a square free detour set such that is independent. Thus . Theorem 3.13 If is a Flower graph then Proof. Let be a Flower graph of order n. Let be the hub, be the vertices on the inner rim and be the vertices at square free detour distance 3 from the hub. Then we have two cases. Case 1. Let be even. Let is the square free detour set such that is independent. Thus Case 2. Let be odd. Then is a square free detour set such that is independent. Hence Theorem 3.14 If is a Helm , then Proof. Let be a Helm of order . Let be any set in . Then we have the following two cases. Case 1. Let be odd. Then is a square free detour set of vertices where is the hub, are the vertices on the rim and are the pendent vertices of such that is independent. Hence Case 2. Let be even. Then is a square free detour set of vertices where is the hub, are the vertices on the rim and are the pendent vertices of such that is independent. Hence Theorem 3.15 If is a Closed Helm , then 313 K. Christy Rani and G. Priscilla Pacifica Proof. Let be a closed Helm of order . Let be the hub, and are the vertices of inner and outer rim of respectively. Then we consider two cases. Case 1. Let be even. Let is the square free detour set such that is independent. Thus Case 2. Let be odd. Then is a square free detour set such that is independent. Hence Theorem 3.16 For any pair of integers and with there exists a connected graph of order with square free detour number and outer independent square free detour number Proof. We consider two cases. Case 1. . Any star with end-vertices has the desired property. Case 2. . Let be a graph obtained from by adding new vertices to Let be the graph derived from by adding new vertices and identifying with Let be the graph derived from by joining the remaining vertices to and of The resulting graph of order is shown in Figure 2. By Theorem 3.3, it is verified that and are the square free detour sets of and so Figure 2: Now, consider the set of all vertices finally added to to obtain It is easy to verify that is the outer independent square free detour set of and so 4. Conclusion In this paper, we determined the outer independent square free detour number of some standard graphs and cycle related graphs. The relationship between the square free detour number and the outer independent square free detour number has been established. Further investigation is open for any other class of graphs. 314 Outer independent square free detour number of a graph References [1] Asir, I. Keerthi, and S. Athisayanathan. Triangle free detour distance in graphs. J. Combin. Math. Combin. Comput 105 (2016). [2] Chartrand, Gary, Garry L. Johns, and Songlin Tian. Detour distance in graphs. Annals of discrete mathematics. Vol. 55. Elsevier (1993): 127-136. [3] Chartrand, Garry L. Johns and Ping Zhang. The detour number of a graph. 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