Ratio Mathematica Volume 47, 2023 Detour Global Domination for Degree Splitting Graphs of Some Graphs C. Jayasekaran* S.V. Ashwin Prakash† Abstract In this paper, we introduced the new concept detour global domina- tion number for degree splitting graph of standard graphs. A set S is called a detour global dominating set of G = (V, E) if S is both detour and global dominating set of G. The detour global domination number is the minimum cardinality of a detour global dominating set in G. Let V (G) be S1 ∪ S2 ∪ · · · ∪ St ∪ T, where Si is the set having at least two vertices of same degree and T = V (G) − ∪Si, where 1 ≤ i ≤ t. The degree splitting graph DS(G) is obtained from G by adding vertices w1, w2, · · · , wt and joining wi to each vertex of Si for i = 1, 2, · · · , t. In this article we recollect the concept of degree splitting graph of a graph and we produced some results based on the detour global domination number for degree splitting graph of path graph, cycle graph, star graph, bistar graph, complete bipartite graph and complete graph. Keywords: Detour set, Dominating set, Detour Domination, Global Domination, Detour Global Domination, Degree Splitting graphs. 2020 AMS subject classifications: 05C12, 05C69.1 *Associate Professor, Department of Mathematics, Pioneer Kumaraswamy College, Nagercoil - 629003, Tamil Nadu, India; jayacpkc@gmail.com. †Research Scholar, Department of Mathematics, Pioneer Kumaraswamy College, Nagercoil - 629003, Tamil Nadu, India; ashwinprakash00@gmail.com. Affliated to Manonmaniam Sundara- nar University, Abishekapatti, Tirunelveli - 627012, Tamil Nadu, India. 1Received on October 10, 2022. Accepted on May 1, 2023. Published online on May 10, 2023. DOI: 10.23755/rm.v39i0.871. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 290 C. Jayasekaran, S.V. Ashwin Prakash 1 Introduction By a graph G = (V, E) we mean a finite, connected, undirected graph with neither loops nor multiple edges. The order |V | and size |E| of G are denoted by p and q respectively. For graph theoretic terminology we refer to West[1]. For vertices x and y in a connected graph G, the detour distance D(x, y) is the length of a longest x − y path in G[2]. An x − y path of length D(x, y) is called an x − y detour. The closed interval ID[x, y] consists of all vertices lying on some x − y detour of G. For S ⊆ V, ID[S] = ∪x,y∈SID[x, y]. A set S of vertices is a detour set if ID[S] = V , and the minimum cardinality of a detour set is the detour number dn(G). A detour set of cardinality dn(G) is called a minimum detour set [3]. A set S ⊆ V (G) in a graph G is a dominating set of G if for every vertex v in V -S, there exists a vertex u ∈ S such that v is adjacent to u. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G[4]. The complement G of a graph G also has V (G) as its point set, but two points are adjacent in G if and only if they are not adjacent in G. A set S ⊆ V (G) is called a global dominating set of G if it is a dominating set of both G and G[5]. A vertex of degree 0 is called an isolated vertex and a vertex of degree 1 is called an end vertex or a pendant vertex. A vertex that is adjacent to a pendant vertex is called a support vertex. Definition 1.1. Let G = (V, E) be a connected graph with atleast two vertices. A set S ⊆ V (G) is said to be a detour global dominating set of G if S is both detour and global dominating set of G. The detour global domination number, denoted by γ̄d(G) is the minimum cardinality of a detour global dominating set of G and the detour global dominating set with cardinality γ̄d(G) is called the γ̄d-set of G or γ̄d(G)-set.[6] In [7], R. Ponraj and S. Somasundaram have initiated a study on degree split- ing graph DS(G) of a graph G which is defined as follows: Definition 1.2. Let G = (V, E) be a graph with V (G) = S1 ∪ S2 ∪ · · · ∪ St ∪ T, where Si is the set having at least two vertices of same degree and T = V (G) − ∪Si, where 1 ≤ i ≤ t. The degree splitting graph DS(G) is obtained from G by adding vertices w1, w2, · · · , wt and joining wi to each vertex of Si for i = 1, 2, · · · , t. Example 1.1. In Figure 1.1, a graph G and its degree splitting graph DS(G) are shown. 291 Detour Global Domination for Degree Splitting Graphs of Some Graphs G DS(G) Figure 1.1 v1 v2 v3 v4 v5 v6 v8 v7 v1 v2 v3 v4 v5 v6 v8 v7 w3 w1 w2 Here, S1 = {v1, v7, v8}, S2 = {v2, v6}, S3 = {v4, v5} and T = {v3}. Remark 1.1. If V (G) = ∪Si, 1 ≤ i ≤ t, then T = ϕ. 2 Some basic results In this section, we recall some basic results of detour global domination num- ber of a graph which will be used throughout the paper. Theorem 2.1. [6] Every isolated vertex of G belongs to every detour global dom- inating set of G. Theorem 2.2. [6] Every full vertex of a connected graph G of order p belongs to every detour global dominating set of G. Theorem 2.3. [6] For the path graph Pp, (p ≥ 4), γ̄d(Pp) = γd(Pp) = ⌈p+23 ⌉. Theorem 2.4. [6] For any star graph K1,n−1, (n ≥ 2), γ̄d(K1,n−1) = n. Theorem 2.5. [6] For any bistar graph Bm,n, m, n ≥ 1, γ̄d(Bm,n) = m + n. Theorem 2.6. [6] For m, n ≥ 2, γ̄d(Km,n) = 2. 292 C. Jayasekaran, S.V. Ashwin Prakash 3 Degree Splitting graphs of known graphs and their Detour Global Domination number Let us find detour global domination number of degree splitting graph DS(G) of the graphs path, cycle, star, bistar, complete bipartite and complete graph. Theorem 3.1. For any integer n ≥ 3, γ̄d(DS(Pn)) = 2. Proof. Let v1v2 · · · vn be the path Pn with partitions S1 = {v2, v3, · · · , vn−1} and S2 = {v1, vn}. To obtain DS(P3) from P3 we add x, which corresponds to S2 also P3 is isomorphic to C4 and to obtain DS(Pn) for n ≥ 4 we add x1 and x2, which corresponds to S1 and S2, respectively. As a result, V (DS(P3)) = {x, v1, v2, v3} and V (DS(Pn)) = {x1, x2, v1, v2, · · · , vn}, where |V (DS(Pn))| = n + 2 for n ≥ 4. Pn DS(Pn) Figure 3.1 v1 v2 v3 vn−1 vn v1 v2 v3 vn−1 vn x2 x1 Consider ID[x, v1] for n = 3, which has only one x − v1 detour path of length 3 that contains all the vertices of DS(P3). As a result, S = {x, v1} is a minimum cardinality detour set. Also, x dominates S2 and v2 is dominated by v1. Thus, S is a minimum detour dominating set of DS(P3). Moreover in DS(P3), x dominates v2 and v3 is dominated by v1. Hence, S is a minimum detour global dominating set of DS(P3) and so γ̄d(DS(P3)) = 2. Consider ID[x1, x2] for n ≥ 4, which has two distinct x1 − x2 detour path of length n that contains all the vertices of DS(Pn). As a result, S = {x1, x2} is a minimum cardinality detour set. Also, x1 dominates S2 and x2 dominates 293 Detour Global Domination for Degree Splitting Graphs of Some Graphs S1. Thus, S is a detour dominating set of DS(Pn). Moreover in DS(Pn), x1 dominates S1 and x2 dominates S2 and hence S is a detour global dominating set DS(Pn). Hence, we conclude that for n ≥ 3, γ̄d(DS(Pn) = 2. Theorem 3.2. For any integer n ≥ 3, γ̄d(DS(Cn)) = 3. Proof. Let v1v2 · · · vn be the cycle Cn. To obtain DS(Cn) for n ≥ 3 we add a vertex x which is adjacent to every vertices in Cn. As a result, V (DS(Cn)) = {x, v1, v2, · · · , vn}, where |V (DS(Cn))| = n + 1 for n ≥ 3. Clearly, DS(Cn) is isomorphic to the wheel graph Wn. Cn DS(Cn) Figure 3.2 vn v1v5 vn v1v5 v2 v3 v4 v2 v3 v4 x Since, DS(Cn) is a connected graph, then γ̄d(DS(Cn)) ≥ 2. Consider the set S = {x, vi}, where 1 ≤ i ≤ n. Clearly, the x − vi detour path of DS(Cn) contains all the vertices of DS(Cn) and also, x dominates every vertices of Cn. Hence, S is a minimum detour dominating set of DS(Cn). But in DS(Cn), the neighbours of vi are not dominated by any vertices of S. So, choose vi+1 from 294 C. Jayasekaran, S.V. Ashwin Prakash DS(Cn) then vi, vi+1 and x dominates every vertices from DS(Cn). Therefore, S1 = S ∪ {vi+1} = {v, vi, vi+1} is a minimum detour global dominating set of DS(Cn) and hence γ̄d(DS(Cn)) = 3. Theorem 3.3. For any integer n ≥ 2, γ̄d(DS(K1,n)) = 2. Proof. Let v1, v2, · · · , vn−1 are the end vertices and v is the full vertex of the star K1,n−1 and x be the corresponding vertex which is added to obtain the graph DS(K1,n). Then V (DS(K1,n)) = {v, v1, v2, · · · , vn, x}. Clearly, |V (DS(K1,n−1)| = n + 2. K1,n DS(K1,n) Figure 3.3 v v1 v2 v3 vn v v1 v2 v3 vn x Since, DS(K1,n) is connected, then 2 ≤ γ̄d(DS(K1,n)) ≤ n + 1. Consider S = {v, vi} for some i, 1 ≤ i ≤ n. Then there are n− detour path which travels between v and vi that includes all the vertices of DS(K1,n). Therefore, S = {v, vi} is a detour set of minimum cardinality. Moreover, v dominates every vi and v itself in DS(K1,n) and vi dominates x in DS(K1,n) for 1 ≤ i ≤ n. Now consider in DS(K1,n) where v dominates x and vi dominates all other vj for 1 ≤ j ̸= i ≤ n. This concludes that S is a minimum detour global dominating set of DS(K1,n) and hence γ̄d(DS(K1,n)) = 2. Theorem 3.4. For the bistar graph Bm,n, γ̄d(DS(Bm,n) = 2. 295 Detour Global Domination for Degree Splitting Graphs of Some Graphs Proof. Consider the bistar graph Bm,n with V (Bm,n) = {u, v, ui, vj/1 ≤ i ≤ m; 1 ≤ j ≤ n}. Here, ui and vj are the vertices adjacent with u and v respec- tively. Let x1 and x2 be the corresponding vertices which are added to obtain DS(Bm,n). Then V (DS(Bm,n)) = {u, v, ui, vj, x1, x2/1 ≤ i ≤ m; 1 ≤ j ≤ n} and so |V (S′(Bm,n)| = m + n + 4. Bm,n DS(Bm,n) Figure 3.4 u v u1 u2 um v1 v2 vn u v u1 u2 um v1 v2 vn x1 x2 Since G is a connected graph, γ̄d(DS(Bm,n)) ≥ 2. Consider ID[x1, x2] which has m + n transversal detour path of length four between x1 and x2 which include all the vertices of DS(Bm,n). Therefore, S = {x1, x2} is a detour set of minimum cardinality. Also, x1 dominates u and v and x2 dominates all ui and vj for 1 ≤ i ≤ n and 1 ≤ j ≤ n in DS(Bm,n). It follows that S is a detour dominating set of DS(Bm,n). Moreover, in DS(Bm,n), x2 dominates x1, u and v; x1 dominates u1, u2, · · · , um, v1, v2, · · · , vm. Clearly, S is a minimum detour global dominating set of DS(Bm,n) and hence γ̄d(DS(Bm,n)) = 2. Theorem 3.5. For any integer m, n ≥ 2, γ̄d(DS(Km,n)) = { 3 if m = n 2 if m ̸= n Proof. Consider Km,n with V (Km,n) = {ui, vj/1 ≤ i ≤ m, 1 ≤ j ≤ n} with partition V1 = {v1, v2, · · · , vm} and V2 = {u1, u2, · · · , un}. Now we consider the following two cases. Case (1) m = n 296 C. Jayasekaran, S.V. Ashwin Prakash In this case each vertex is of same degree and so let x be the added vertex which is adjacent to every ui and vj, 1 ≤ i ≤ m and 1 ≤ j ≤ n. Thus, we obtain the graph DS(Km,n). Then V (DS(Km,n)) = {ui, vj, x/1 ≤ i ≤ m; 1 ≤ j ≤ n} and so |V (DS(Km,n))| = m + n + 1. DS(Km,m) Figure 3.5 v1 u1 unu2 u3 v2 v3 vm x Consider the set S = {vi, uj} for some i, j where 1 ≤ i ≤ m and 1 ≤ j ≤ n. Then ID[vi, uj] = V (DS(Km,n)) and also vi dominates every uj and x; uj dominates every vi. Hence, S = {vi, uj} is a minimum detour dominating set. Now consider DS(Km,n), where x is an isolated vertex and not dominated by any vertex of S. This shows that S is not a detour global dominating set of DS(Km,n). Hence, we include x in S such that S = {ui, vj, x} is a detour global dominating set of DS(Km,n) for 1 ≤ i ≤ m; 1 ≤ j ≤ n. Therefore, for m = n, γ̄d(DS(Km,n)) = 3. Case (2) m ̸= n In this case each vertex ui is of same degree and each vertex vj is of same degree where deg(ui) ̸= deg(vj), 1 ≤ i ≤ m; 1 ≤ j ≤ n so let x1 and x2 be the added vertex where x1 is adjacent to every ui and x2 is adjacent to every vj. Thus, we obtain the graph DS(Km,n). Then V (DS(Km,n)) = {ui, vj, x1, x2/1 ≤ i ≤ m; 1 ≤ j ≤ n} and so |V (DS(Km,n))| = m + n + 2. 297 Detour Global Domination for Degree Splitting Graphs of Some Graphs DS(Km,m) Figure 3.6 v1 u1 un u2 u3 v2 v3 vm x1 x2 Consider the set S = {x1, x2}, where ID[x1, x2] = V (DS(Km,n)) and also x1 dominates every vi and x2 dominates every uj. Hence, S = {x1, x2} is a minimum detour dominating set. Now consider DS(Km,n), where x1 dominates every ui for 1 ≤ i ≤ n and x2 dominates every vj for 1 ≤ j ≤ n. Clearly, S = {x1, x2} is a minimum detour global dominating set and so for m ̸= n, γ̄d(DS(Km,n)) = 2. Theorem 3.6. For any integer n ≥ 2, γ̄d(DS(Kn)) = n. Proof. Here, DS(Kn) is isomorphic to Kn+1. We know that all the vertices are isolated vertices in the complement graph of DS(Kn). Therefore, the detour global dominating set must contain all the vertices of DS(Kn) and so, for n ≥ 2, γ̄d(DS(Kn)) = n. 4 Conclusion Inspired by the global dominating set and detour set we introduce the detour global dominating set for degree splitting graph. We have determined the de- tour global domination number for degree splitting graph of path graph, cycle graph, star graph, bistar graph, complete bipartite graph and complete graph. Fur- thermore our results are also justified with suitable examples. The detour global domination number can also be obtained for many more graphs. 298 C. Jayasekaran, S.V. Ashwin Prakash Acknowledgements The authors express their gratitude to the management- Ratio Mathematica for their constant support towards the successful completion of this work. We wish to thank the anonymous reviewers for the valuable suggestions and comments. References [1] D.B. West. 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